Congruent MP3: Construct viable arguments and critique the reasoning - - PDF document

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Geometry

Congruent Triangles

2015-10-23 www.njctl.org

Slide 3 / 183 Table of Contents

· Congruent Triangles · Isosceles Triangle Theorem · SSS Congruence · SAS Congruence · ASA Congruence · AAS Congruence · HL Congruence · CPCTC · Triangle Congruence Proofs

click on the topic to go to that section

· Proving Congruence · PARCC Sample Questions

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Throughout this unit, the Standards for Mathematical Practice are used. MP1: Making sense of problems & persevere in solving them. MP2: Reason abstractly & quantitatively. MP3: Construct viable arguments and critique the reasoning of

• thers.

MP4: Model with mathematics. MP5: Use appropriate tools strategically. MP6: Attend to precision. MP7: Look for & make use of structure. MP8: Look for & express regularity in repeated reasoning. Additional questions are included on the slides using the "Math Practice" Pull-tabs (e.g. a blank one is shown to the right on this slide) with a reference to the standards used. If questions already exist on a slide, then the specific MPs that the questions address are listed in the Pull-tab.

Slide 4 (Answer) / 183

Throughout this unit, the Standards for Mathematical Practice are used. MP1: Making sense of problems & persevere in solving them. MP2: Reason abstractly & quantitatively. MP3: Construct viable arguments and critique the reasoning of

• thers.

MP4: Model with mathematics. MP5: Use appropriate tools strategically. MP6: Attend to precision. MP7: Look for & make use of structure. MP8: Look for & express regularity in repeated reasoning. Additional questions are included on the slides using the "Math Practice" Pull-tabs (e.g. a blank one is shown to the right on this slide) with a reference to the standards used. If questions already exist on a slide, then the specific MPs that the questions address are listed in the Pull-tab.

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Math Practice

Slide 5 / 183

Congruent Triangles

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Slide 6 / 183 Similar Triangles

We learned in the Similar Triangles topic (Triangles unit) that if two triangles are similar: · All their angles are congruent · All their corresponding sides are in proportion We also learned how to identify the corresponding sides as being opposite to equal angles, or subtended by equal angles And we learned that the constant of proportionality for the corresponding sides of one triangle to the other was called "k." If needed, go back to review that topic before proceeding.

Slide 7 / 183 Congruent Triangles

Congruent triangles are a special case of similar triangles. The constant of proportionality is one, so the corresponding sides are of equal measure. For congruent triangles, all the angles are congruent AND all the corresponding sides are congruent.

Slide 8 / 183 Naming Congruent Triangles

Just as in the case of similar triangles, the naming of congruent triangles is important: order matters. The statement: ΔABC is congruent to ΔDEF indicates that these triangles are congruent. AND that these angle measures are equal: m∠A = m∠D m∠B = m∠E m∠C = m∠F AND these lengths are equal: AB = DE BC = EF CA = FD

Slide 8 (Answer) / 183 Naming Congruent Triangles

Just as in the case of similar triangles, the naming of congruent triangles is important: order matters. The statement: ΔABC is congruent to ΔDEF indicates that these triangles are congruent. AND that these angle measures are equal: m∠A = m∠D m∠B = m∠E m∠C = m∠F AND these lengths are equal: AB = DE BC = EF CA = FD

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Math Practice

MP6 Make sure that the students understand that the order in which you name the triangles matters. This slide explains why.

Slide 9 / 183 Proving Triangles Congruent

We can prove triangles congruent by proving the measures of all three corresponding angles and the lengths of all three corresponding sides are equal. Earlier we showed that we need to prove only two angles are congruent to show that triangles are similar, since the third angle must then be congruent. There are similar shortcuts to proving triangles congruent.

Slide 10 / 183 Third Angle Theorem

Recall the proof showing if we know that two pairs of corresponding angles are congruent, then the third pair of corresponding angles are congruent as well. Statement Reason 1 ∠A ≅ ∠D and ∠B ≅ ∠E Given 2 m∠A = m∠D; m∠B = m∠E Definition of ≅ angles 3 m∠A+ m∠B + m∠C = 180º m∠D+ m∠E + m∠F = 180º Triangle Sum Theorem 4 m∠D+ m∠E + m∠C = 180º m∠D+ m∠E + m∠F = 180º Substitution Property of Equality 5 m∠D + m∠E + m∠C = m∠D + m∠E + m∠F Substitution Property of Equality 6 m∠C = m∠F Subtraction Property of Equality

Slide 11 / 183 Corresponding Parts

Let's review identifying the corresponding parts (angles and sides) of pairs of triangles.

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Given that ΔABC is congruent to ΔDEF, identify all the congruent corresponding parts

A B C D E F

Corresponding Parts Slide 12 (Answer) / 183

Given that ΔABC is congruent to ΔDEF, identify all the congruent corresponding parts

A B C D E F

Corresponding Parts

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Answer ∠A ≅ ∠D ∠B ≅ ∠E ∠C ≅ ∠F AB ≅ DE BC ≅ EF CA ≅ FD This example addresses MP6.

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A B C D E F

Given that ΔABC is congruent to ΔDEF, the triangles are marked accordingly in this diagram.

Corresponding Parts Slide 14 / 183

A B D C E

Part Corresponding Part Segment AB Segment ED ∠A ∠E Segment AC Segment EC ∠B ∠D Segment CB Segment CD ∠ACB ∠ECD

ΔABC ≅ ΔEDC

Slide 15 / 183 Example

Corresponding Sides Corresponding Angles Given that ΔABC ≅ ΔLMN, identify all the corresponding angles and

• sides. (Draw a diagram)

Slide 15 (Answer) / 183 Example

Corresponding Sides Corresponding Angles Given that ΔABC ≅ ΔLMN, identify all the corresponding angles and

• sides. (Draw a diagram)

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Teacher Notes

Corresponding Sides AB LM BC MN AC LN = ~ = ~ = ~ Corresponding Angles Have students arrive at the answers as a class, or independently. This example addresses MP5 & MP6 Additional Q's that address MP standards: How could you start this problem? (MP1) How could you use a drawing to assist with this problem? (MP5) How can you make sure that your answer is accurate? (MP6)

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1 What is the corresponding part to ∠ J ? A ∠R B ∠K C ∠Q D ∠P J K L R Q P ΔJKL ≅ ΔPQR

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1 What is the corresponding part to ∠ J ? A ∠R B ∠K C ∠Q D ∠P J K L R Q P ΔJKL ≅ ΔPQR

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Answer

D Slide 17 / 183

2 What is the corresponding part to ∠Q? A ∠R B ∠K C ∠Q D ∠P J K L R Q P ΔJKL ≅ ΔPQR

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2 What is the corresponding part to ∠Q? A ∠R B ∠K C ∠Q D ∠P J K L R Q P ΔJKL ≅ ΔPQR

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Answer

B Slide 18 / 183

3 What is the corresponding part to QP? A JL B LK C KJ D PQ J K L R Q P ΔJKL ≅ ΔPQR

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3 What is the corresponding part to QP? A JL B LK C KJ D PQ J K L R Q P ΔJKL ≅ ΔPQR

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Answer

C Slide 19 / 183

Z X C V B 4 The congruence statement for the two triangles is: A ΔBVC ≅ ΔXCZ B ΔXCB ≅ ΔBCX C ΔVBC ≅ ΔZXC D ΔCBV ≅ ΔCZX

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Z X C V B 4 The congruence statement for the two triangles is: A ΔBVC ≅ ΔXCZ B ΔXCB ≅ ΔBCX C ΔVBC ≅ ΔZXC D ΔCBV ≅ ΔCZX

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Answer

C Slide 20 / 183

5 Complete the congruence statement: ΔXYZ ≅ A ΔXWZ B ΔZWX C ΔWXZ D ΔZXW Y Z W X

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5 Complete the congruence statement: ΔXYZ ≅ A ΔXWZ B ΔZWX C ΔWXZ D ΔZXW Y Z W X

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Answer

B Slide 21 / 183 Properties of Congruence and Equality

We will be using the three properties of congruence we learned earlier Reflexive Property of Congruence Symmetric Property of Congruence Transitive Property of Congruence As well as the four properties of equality we learned earlier Reflexive Property of Equality Symmetric Property of Equality Transitive Property of Equality Substitution Property of Equality

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Proving Congruence SSS (Side-Side-Side)

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Slide 23 / 183

Congruent triangles have all congruent sides and angles. However, congruence can be proven by showing less than that. We will prove some theorems which you can then use as shortcuts to proving two triangles congruent. It is not necessary to prove that all the angles and sides are congruent.

Proving Congruence Slide 24 / 183 Side-Side-Side Triangle Congruence

If two triangles have the two sides equal to two sides respectively, and also have the base equal to the base, then they also have the angles equal which are contained by the equal straight lines. Euclid - Book 1: Proposition 8

Slide 25 / 183 Side-Side-Side Triangle Congruence

Euclid showed that: Having three equal sides requires having three equal angles. Therefore, having three pairs of equal sides verifies that two triangles are congruent since all their corresponding sides and angles must be congruent.

Slide 26 / 183 Side-Side-Side Triangle Congruence

Euclid's argument of this (and for some of the following postulates/ theorems) was based on transposing one triangle on top of the

• ther.

He confirmed that if all the corresponding sides are equal, once you place one triangle atop the other in the correct orientation, all the sides have to line up and all the angles must as well. Recall the Fourth Axiom: Things which coincide with

• ne another are equal to
• ne another.

Slide 27 / 183 Side-Side-Side Triangle Congruence

Click here to go to the lab titled, "Triangle Congruence SSS"

This is shown below. Can you imagine a way that the corresponding angles could be

• f different measure without changing the length of one of the

sides? This is often called the SSS Triangle Congruence for short.

Slide 27 (Answer) / 183 Side-Side-Side Triangle Congruence

Click here to go to the lab titled, "Triangle Congruence SSS"

This is shown below. Can you imagine a way that the corresponding angles could be

• f different measure without changing the length of one of the

sides? This is often called the SSS Triangle Congruence for short.

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Math Practice

The Lab - Triangle Congruence SSS addresses MP3, MP4, MP5, MP6, MP7 & MP8

Slide 28 / 183 Example 1

A F K B G

Solution: The congruence marks on the sides show that each of the three sides in one triangle is congruent with that of the other. By SSS, this proves congruence. Please note that this requires that all three sides are congruent. Prove that ΔAFK is congruent to ΔBGK

Slide 28 (Answer) / 183 Example 1

A F K B G

Solution: The congruence marks on the sides show that each of the three sides in one triangle is congruent with that of the other. By SSS, this proves congruence. Please note that this requires that all three sides are congruent. Prove that ΔAFK is congruent to ΔBGK

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Math Practice

This example addresses MP3 & MP6

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F G H K J

Example 2

Given: FG = JK, FH = JH, and H is the midpoint of GK Prove: ΔFGH ≅ ΔJKH

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F G H K J

Example 2

Given: FG = JK, FH = JH, and H is the midpoint of GK Prove: ΔFGH ≅ ΔJKH

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Math Practice

This example addresses MP3 & MP6 Additional Q's to address MP Standards: What information are we given? (MP1) What do we need to prove? (MP1) If segments have equal length, what else can we say about them? Why? (MP6 & MP3) What does the midpoint mean? (MP6) How is midpoint related to congruent segments? (MP7)

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Statement Reason 1 FG = JK, FH = JH and H is the midpoint of GK Given 2 FG ≅ JK, FH ≅ JH Definition of congruent segments 3 GH ≅ HK Definition of midpoint 4 ΔFGH ≅ ΔJKH Side-Side-Side Triangle Congruence

G F H J K

Example 2

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A B C H J K 6 ΔABC ≅ ΔHJK True False

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A B C H J K 6 ΔABC ≅ ΔHJK True False

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Answer

True Slide 32 / 183

A B C H J K 7 ΔCAB ≅ ΔHJK True False

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A B C H J K 7 ΔCAB ≅ ΔHJK True False

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Answer False The order of the letters in the triangle name makes a difference.

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R S T U 8 ΔSRT ≅ ΔSUT True False

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R S T U 8 ΔSRT ≅ ΔSUT True False

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Answer

True

SSS Triangle congruence applies because ST ≅ ST ; Reflexive Property

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9 Provide the reason for the second step.

Statement Reason 1 RS ≅ US, RT ≅ UT Given 2 ST ≅ ST ? 3 ΔSRT ≅ ΔSUT Side-Side-Side Triangle Congruence

R S T U A Given B Side-Side-Side Triangle Congruence C Reflexive property of congruence D Substitution property of congruence E Transitive property of congruence

Answer

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10 ΔABC is congruent to A ΔQRS B ΔSRQ C ΔACB D ΔRSQ A B C Q R S 3 4 5 3 4 5

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10 ΔABC is congruent to A ΔQRS B ΔSRQ C ΔACB D ΔRSQ A B C Q R S 3 4 5 3 4 5

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Answer

B Slide 36 / 183

Proving Congruence SAS (Side-Angle-Side)

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Slide 37 / 183 Side-Angle-Side Triangle Congruence

If two triangles have the two sides equal to two sides respectively, and have the angles contained by the equal straight lines equal, they will also have the base equal to the base, the triangle will be equal to the triangle and the remaining angles will be equal to the remaining angles respectively, namely those which the equal sides subtend.

• Euclid's Elements - Book One: Proposition 4

Slide 38 / 183 Side-Angle-Side Triangle Congruence

As in Side-Side-Side Triangle Congruence, Euclid verifies Side-Angle-Side Triangle Congruence by superposition (transposing one triangle atop the other.) He thereby indicates that if two sides of two triangles, and the angles contained by those sides, are equal, then all of the sides and angles must be equal...showing congruence.

Slide 39 / 183

Given that two triangles have two equal corresponding sides and equal angles contained by those two equal sides, the third sides must also be equal. Tap below and the third side of each triangle will become visible.

Side-Angle-Side Triangle Congruence

Tap to reveal third side of the triangles

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It is clear that the third side of each triangle is completely defined by the two other sides and their included angle. So, the third sides must also be congruent. This is often called the SAS Triangle Congruence for short.

Side-Angle-Side Triangle Congruence Slide 41 / 183

So, if you can show that two triangles have two sides as well as the included angle (the angle formed by the two equal sides) to be equal, then all the sides and angles are congruent and the triangles are congruent.

Side-Angle-Side Triangle Congruence

Click here to go to the lab titled, "Triangle Congruence SAS"

Slide 41 (Answer) / 183

So, if you can show that two triangles have two sides as well as the included angle (the angle formed by the two equal sides) to be equal, then all the sides and angles are congruent and the triangles are congruent.

Side-Angle-Side Triangle Congruence

Click here to go to the lab titled, "Triangle Congruence SAS"

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Math Practice

The Lab - Triangle Congruence SAS addresses MP3, MP4, MP5, MP6, MP7 & MP8

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1 2 L M P N O

Example

Given: MP ≅ NP and LP ≅ OP Prove: ΔMLP ≅ ΔNOP

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1 2 L M P N O

Example

Given: MP ≅ NP and LP ≅ OP Prove: ΔMLP ≅ ΔNOP

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Math Practice

This example (slides #42-44) addresses MP3 & MP6 Additional Q's to address MP Standards: What information are we given? (MP1) What do we need to prove? (MP1) How are these angles related? Why? (MP6 & MP3)

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11 Provide the reason for line 2.

1 2 L M P N O

Given: MP ≅ NP and LP ≅ OP Prove: ΔMLP ≅ ΔNOP

Statement Reason 1 MP ≅ NP and LP ≅ OP Given 2 ∠1 ≅ ∠2 ? 3 ΔMLP ≅ ΔNOP ?

A Given B Side-Side-Side Triangle Congruence C Side-Angle-Side Triangle Congruence D Vertical angles are congruent E Alternate interior angles are congruent

Answer

Slide 44 / 183

12 Provide the reason for line 3.

Statement Reason 1 MP ≅ NP and LP ≅ OP Given 2 ∠1 ≅ ∠2 ? 3 ΔMLP ≅ ΔNOP ?

A Given B Side-Side-Side Triangle Congruence C Side-Angle-Side Triangle Congruence D Vertical angles are congruent E Alternate interior angles are congruent

Given: MP ≅ NP and LP ≅ OP Prove: ΔMLP ≅ ΔNOP

1 2 L M P N O

Answer

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13 What is the included angle of the given sides of the triangle? A ∠J B ∠K C ∠L

Hint: Draw the triangle!

ΔJKL, sides KL and JK

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13 What is the included angle of the given sides of the triangle? A ∠J B ∠K C ∠L

Hint: Draw the triangle!

ΔJKL, sides KL and JK

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Answer

B Slide 46 / 183

P Q R T V

4 4 5 5

100° 100° S 14 List the congruent parts of the triangles below. Is ΔPQR ≅ ΔSTV? Yes No

Slide 46 (Answer) / 183

P Q R T V

4 4 5 5

100° 100° S 14 List the congruent parts of the triangles below. Is ΔPQR ≅ ΔSTV? Yes No

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Answer

Yes, by SAS Triangle Congruence

PQ ≅ ST ∠Q ≅ ∠T RQ ≅ VT

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F G H X Y Z 46° 46°

10 10 7 7

Why? 15 Is ΔFGH ≅ ΔXYZ by Side-Angle-Side Triangle Congruence? Yes No

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F G H X Y Z 46° 46°

10 10 7 7

Why? 15 Is ΔFGH ≅ ΔXYZ by Side-Angle-Side Triangle Congruence? Yes No

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Answer

No, the angles marked congruent are not the included angles.

FG ≅ XY ∠H ≅ ∠Z HF ≅ ZX

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A B C D 16 Using SAS Triangle Congruence, what information is needed to show ΔABC ≅ ΔDCB ? A ∠DBC ≅ ∠ABC B ∠B ≅ ∠C C ∠ABD ≅ ∠DCA D ∠ABC ≅ ∠DCB

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A B C D 16 Using SAS Triangle Congruence, what information is needed to show ΔABC ≅ ΔDCB ? A ∠DBC ≅ ∠ABC B ∠B ≅ ∠C C ∠ABD ≅ ∠DCA D ∠ABC ≅ ∠DCB

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Answer

D Slide 49 / 183

17 What type of congruence exists between the two triangles? A SSS Triangle Congruence B SAS Triangle Congruence C Not enough information

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17 What type of congruence exists between the two triangles? A SSS Triangle Congruence B SAS Triangle Congruence C Not enough information

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Answer

B

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18 What type of congruence exists between the two triangles? A SSS Triangle Congruence B SAS Triangle Congruence C Not enough information

Slide 50 (Answer) / 183

18 What type of congruence exists between the two triangles? A SSS Triangle Congruence B SAS Triangle Congruence C Not enough information

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Answer

B Slide 51 / 183

19 What type of congruence exists between the two triangles? A SSS Triangle Congruence B SAS Triangle Congruence C Not enough information

Slide 51 (Answer) / 183

19 What type of congruence exists between the two triangles? A SSS Triangle Congruence B SAS Triangle Congruence C Not enough information

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Answer

A Slide 52 / 183

20 What type of congruence exists between the two triangles? A SSS Triangle Congruence B SAS Triangle Congruence C Not enough information

Slide 52 (Answer) / 183

20 What type of congruence exists between the two triangles? A SSS Triangle Congruence B SAS Triangle Congruence C Not enough information

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Answer

C

Slide 53 / 183

21 What type of congruence exists between the two triangles? A SSS Triangle Congruence B SAS Triangle Congruence C Not enough information

Slide 53 (Answer) / 183

21 What type of congruence exists between the two triangles? A SSS Triangle Congruence B SAS Triangle Congruence C Not enough information

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Answer

C Slide 54 / 183

22 What type of congruence exists between the two triangles? A SSS Triangle Congruence B SAS Triangle Congruence C Not enough information 45° 45° 12 12 A B C D

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22 What type of congruence exists between the two triangles? A SSS Triangle Congruence B SAS Triangle Congruence C Not enough information 45° 45° 12 12 A B C D

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Answer

B Slide 55 / 183

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Proving Congruence ASA (Angle-Side-Angle)

Slide 56 / 183 Angle-Side-Angle Triangle Congruence

Another way to prove two triangles are congruent makes use of Euclid's Fifth Postulate. This illustration should look familiar from the unit on parallel

• lines. It shows non-parallel lines intersected by a transversal.

1 2 3 4

Slide 57 / 183

We know from Euclid's Fifth Postulate, that the non-parallel lines will intersect on the side of the transversal on which the sum of the interior angles is less than 180º. In this case, that's the side of the transversal with angles 2 and 4. 1 2 3 4

Angle-Side-Angle Triangle Congruence Slide 58 / 183

By extending the lines and decreasing angles 2 and 4, we can see where the non-parallel lines intersect. This forms a triangle in which the transversal is one side and the two non-parallel lines form the other two sides. 1 2 3 4

Angle-Side-Angle Triangle Congruence Slide 59 / 183

You can see that we have formed a triangle on the right side of the transversal, with the transversal providing one side and the two non-parallel lines the other two sides. Let's examine that triangle. 1 2 3 4 5

A C B

Angle-Side-Angle Triangle Congruence Slide 60 / 183

We can see that ∠2 and ∠4 lead to only one possible value for ∠5, since the angles must add to 180º. That means that two triangles with two corresponding angles which are congruent, must have their third angles equal, so they are similar...but we knew that from earlier. 1 2 3 4 5

A C B

Angle-Side-Angle Triangle Congruence Slide 61 / 183

Now, we also know they have corresponding sides between the two given angles, which are congruent. That means they are not only the same shape, but also the same size. The two triangles must be congruent. This is called ASA Triangle Congruence for short. 1 2 3 4 5

A C B

Angle-Side-Angle Triangle Congruence

To see a visual representation of what we discussed for ASA Triangle Congruence, click the link below. http://www.mathopenref.com/congruentasa.html

Click here to go to the lab titled, "Triangle Congruence ASA"

Slide 61 (Answer) / 183

Now, we also know they have corresponding sides between the two given angles, which are congruent. That means they are not only the same shape, but also the same size. The two triangles must be congruent. This is called ASA Triangle Congruence for short. 1 2 3 4 5

A C B

Angle-Side-Angle Triangle Congruence

To see a visual representation of what we discussed for ASA Triangle Congruence, click the link below. http://www.mathopenref.com/congruentasa.html

Click here to go to the lab titled, "Triangle Congruence ASA"

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Math Practice

The Lab - Triangle Congruence ASA addresses MP3, MP4, MP5, MP6, MP7 & MP8

Slide 62 / 183

W X Y 23 What is the included side between ∠X and ∠W? A YX B YW C XW

Slide 62 (Answer) / 183

W X Y 23 What is the included side between ∠X and ∠W? A YX B YW C XW

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Answer

C Slide 63 / 183

W X Y 24 What is the included side between ∠X and ∠Y ? A XW B YX C YW

Slide 63 (Answer) / 183

W X Y 24 What is the included side between ∠X and ∠Y ? A XW B YX C YW

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Answer

B Slide 64 / 183

M N O P 25 What information is needed to have ASA Triangle Congruence between the two triangles? A B C D

Slide 64 (Answer) / 183

M N O P 25 What information is needed to have ASA Triangle Congruence between the two triangles? A B C D

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Answer

C

NP = PN is true by the Reflexive Property. If you mark the diagram to show the corresponding congruent parts, you can see NP = PN is needed ~ ~

Slide 65 / 183

A C D B 26 What information is needed to have ASA Triangle Congruence between the two triangles? A B C D

Slide 65 (Answer) / 183

A C D B 26 What information is needed to have ASA Triangle Congruence between the two triangles? A B C D

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Answer

B

is true by the Reflexive Property.

Slide 66 / 183

E F G M H 27 Why is ∠FME ≅ ∠GMH? A ASA Triangle Congruence B vertical angles C included angles D congruent

Slide 66 (Answer) / 183

E F G M H 27 Why is ∠FME ≅ ∠GMH? A ASA Triangle Congruence B vertical angles C included angles D congruent

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Answer

B

Mark the congruent vertical angles when you see two lines intersect

Slide 67 / 183

28 What type of congruence exists between the two triangles? A SSS Triangle Congruence B SAS Triangle Congruence C ASA Triangle Congruence D Not enough information Q R U T S

Slide 67 (Answer) / 183

28 What type of congruence exists between the two triangles? A SSS Triangle Congruence B SAS Triangle Congruence C ASA Triangle Congruence D Not enough information Q R U T S

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Answer

A

Slide 68 / 183

When you have overlapping figures that share sides and/or angles, marking the diagram with the given information and separating the triangles (when needed) make it easier to understand the problem. Another strategy that you could use is to look for repeating letters once you separate the two triangles. When 2 letters repeat, then you have a common side shared. When 1 letter repeats, then you have a common angle shared.

Strategy to Prove Congruence Slide 69 / 183

29 What type of congruence exists between ΔJLM and ΔNLK? A SSS Triangle Congruence B SAS Triangle Congruence C ASA Triangle Congruence D Not enough information J M N K L

Pull the triangles apart! Mark the congruent parts! Are there any common sides/angles (look for letters that repeat)?

Hints:

click to reveal click to reveal click to reveal

Slide 69 (Answer) / 183

29 What type of congruence exists between ΔJLM and ΔNLK? A SSS Triangle Congruence B SAS Triangle Congruence C ASA Triangle Congruence D Not enough information J M N K L

Pull the triangles apart! Mark the congruent parts! Are there any common sides/angles (look for letters that repeat)?

Hints:

click to reveal click to reveal click to reveal

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Answer

B Slide 70 / 183

A B C Q R 30 What type of congruence exists between ΔABQ and ΔCBR? A SSS Triangle Congruence B SAS Triangle Congruence C ASA Triangle Congruence D Not enough information

Mark the diagram with the given information. Be careful you don't always use all information Hint

click to reveal

Slide 70 (Answer) / 183

A B C Q R 30 What type of congruence exists between ΔABQ and ΔCBR? A SSS Triangle Congruence B SAS Triangle Congruence C ASA Triangle Congruence D Not enough information

Mark the diagram with the given information. Be careful you don't always use all information Hint

click to reveal

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Answer

C Slide 71 / 183

C B Q R A B R 31 What type of congruence exists between ΔQAR and ΔRCQ? A SSS Triangle Congruence B SAS Triangle Congruence C ASA Triangle Congruence D Not enough information

Pull the triangles apart! Mark the congruent parts! Are there any common sides/angles (look for letters that repeat)? Hints:

click to reveal click to reveal click to reveal

Slide 71 (Answer) / 183

C B Q R A B R 31 What type of congruence exists between ΔQAR and ΔRCQ? A SSS Triangle Congruence B SAS Triangle Congruence C ASA Triangle Congruence D Not enough information

Pull the triangles apart! Mark the congruent parts! Are there any common sides/angles (look for letters that repeat)? Hints:

click to reveal click to reveal click to reveal

[This object is a pull tab]

Answer

A

Slide 72 / 183

S T N D A

vertical

32 What type of congruence exists between ΔSAN and ΔDAT? A SSS Triangle Congruence B SAS Triangle Congruence C ASA Triangle Congruence D Not enough information At the intersection of two lines you always have _____ angles. Hint

Click to Reveal Click

Given: SA

≅ DA

AN

≅ AT

Slide 72 (Answer) / 183

S T N D A

vertical

32 What type of congruence exists between ΔSAN and ΔDAT? A SSS Triangle Congruence B SAS Triangle Congruence C ASA Triangle Congruence D Not enough information At the intersection of two lines you always have _____ angles. Hint

Click to Reveal Click

Given: SA

≅ DA

AN

≅ AT

[This object is a pull tab]

Answer

B

Slide 73 / 183

33 What type of congruence exists between the two triangles? A SSS Triangle Congruence B SAS Triangle Congruence C ASA Triangle Congruence D Not enough information

Slide 73 (Answer) / 183

33 What type of congruence exists between the two triangles? A SSS Triangle Congruence B SAS Triangle Congruence C ASA Triangle Congruence D Not enough information

[This object is a pull tab]

Answer D There are two Sides and

• ne Angle congruent, but

they are not in the correct

• rder.

Slide 74 / 183

34 What type of congruence exists between the two triangles?

A

SSS Triangle Congruence

B

SAS Triangle Congruence

C

ASA Triangle Congruence

D

Not enough information C P M S A

Hint: Mark the given information into your diagram. Identifying vertical angles plays an important part.

click to reveal

Given: PA

≅ MA ∠P ≅ ∠M

Slide 74 (Answer) / 183

34 What type of congruence exists between the two triangles?

A

SSS Triangle Congruence

B

SAS Triangle Congruence

C

ASA Triangle Congruence

D

Not enough information C P M S A

Hint: Mark the given information into your diagram. Identifying vertical angles plays an important part.

click to reveal

Given: PA

≅ MA ∠P ≅ ∠M

[This object is a pull tab]

Answer C

Slide 75 / 183

Return to Table

• f Contents

Proving Congruence AAS (Angle-Angle-Side)

Slide 76 / 183

Based on that same logic, if ANY two corresponding angles and

• ne corresponding side of a pair of triangles are congruent, the

triangles must also be congruent. This follows from the fact that the Triangle Sum Theorem tells us that once we know the measures of two angles, we know the measure of the third, since they must add to 180º. 1 2 3 4 5

A C B

Angle-Angle-Side Triangle Congruence Slide 77 / 183

Another way of looking at these two theorems is that once you show that two corresponding angles in two triangles are congruent, you know that the third angles are congruent and that the two triangles are similar. That means that they have the same shape. If you can show a side in one of those triangles is congruent to the corresponding side of the other, you know that they are same size. Thus the scale factor, k, is 1. If they are the same size and shape, they are congruent.

ASA and AAS Triangle Congruence Slide 78 / 183

It is really just a formality whether you use the term ASA or AAS, since all three angles must be congruent. However, to note the difference, if the angles are both adjacent to the side which has shown to be congruent, the reason for congruence is ASA (∆ABC ≅ ∆DEF). If not, it is AAS (∆GHI ≅ ∆JKL).

ASA and AAS Triangle Congruence

A B C D E F G H I J K L VS. ASA AAS

Slide 79 / 183 Example

C A H T Given: ∠H ≅ ∠C

∠HTA ≅ ∠CTA

Is ΔCTA ≅ ΔHTA ?

Slide 79 (Answer) / 183 Example

C A H T Given: ∠H ≅ ∠C

∠HTA ≅ ∠CTA

Is ΔCTA ≅ ΔHTA ?

[This object is a pull tab]

Math Practice

This example (and the next two slides) addresses MP3 & MP6 Additional Q's to address MP Standards: What information are we given? (MP1) What do we need to prove? (MP1) What properties/theorems/postulates might help you? (MP1) How are the triangles related? Why? (MP6 & MP3)

• Ans: looking for "adjacent"; sharing the

common side AT, for reflexive property

Slide 80 / 183 Example

1) Mark the diagram: C A H T

Given: ∠H ≅ ∠C

∠HTA ≅ ∠CTA

Slide 81 / 183 Example

2) By the reflexive property:

Therefore, ΔHTA ≅ ΔCTA by AAS Triangle Congruence. AT ≅ AT

C A H T congruence statement?

Slide 82 / 183

Given: 35 What type of congruence exists, if any, between the two triangles? A SSS Triangle Congruence B SAS Triangle Congruence C ASA Triangle Congruence D AAS Triangle Congruence E Not enough information D E F H G

Slide 82 (Answer) / 183

Given: 35 What type of congruence exists, if any, between the two triangles? A SSS Triangle Congruence B SAS Triangle Congruence C ASA Triangle Congruence D AAS Triangle Congruence E Not enough information D E F H G

[This object is a pull tab]

Answer

D

What is the congruence statement?

Slide 83 / 183

36 What type of congruence exists, if any, between the two triangles? A SSS Triangle Congruence B SAS Triangle Congruence C ASA Triangle Congruence D AAS Triangle Congruence E Not enough information A B C Q R S

Slide 83 (Answer) / 183

36 What type of congruence exists, if any, between the two triangles? A SSS Triangle Congruence B SAS Triangle Congruence C ASA Triangle Congruence D AAS Triangle Congruence E Not enough information A B C Q R S

[This object is a pull tab]

Answer E Two angles congruent is not enough information.

Slide 84 / 183

37 What type of congruence exists, if any, between the two triangles? A SSS Triangle Congruence B SAS Triangle Congruence C ASA Triangle Congruence D AAS Triangle Congruence E Not enough information

Slide 84 (Answer) / 183

37 What type of congruence exists, if any, between the two triangles? A SSS Triangle Congruence B SAS Triangle Congruence C ASA Triangle Congruence D AAS Triangle Congruence E Not enough information

[This object is a pull tab]

Answer C ASA Triangle Congruence The two triangles share a common side which is congruent via the Reflexive Property.

Slide 85 / 183

Q W E R T 38 What type of congruence exists, if any, between the two triangles? A SSS Triangle Congruence B SAS Triangle Congruence C ASA Triangle Congruence D AAS Triangle Congruence E Not enough information

Slide 85 (Answer) / 183

Q W E R T 38 What type of congruence exists, if any, between the two triangles? A SSS Triangle Congruence B SAS Triangle Congruence C ASA Triangle Congruence D AAS Triangle Congruence E Not enough information

[This object is a pull tab]

Answer D AAS The vertical angles are congruent. The congruent side is nonincluded so it cannot be ASA.

Slide 86 / 183

A S D F G H 39 What type of congruence exists, if any, between the two triangles? A SSS Triangle Congruence B SAS Triangle Congruence C ASA Triangle Congruence D AAS Triangle Congruence E Not enough information

Slide 86 (Answer) / 183

A S D F G H 39 What type of congruence exists, if any, between the two triangles? A SSS Triangle Congruence B SAS Triangle Congruence C ASA Triangle Congruence D AAS Triangle Congruence E Not enough information

[This object is a pull tab]

Answer B SAS Imagine you are walking around the figures, you must encounter the congruent parts in the correct order to use SAS congruence.

Slide 87 / 183

40 What type of congruence exists, if any, between the two triangles? A SSS Triangle Congruence B SAS Triangle Congruence C ASA Triangle Congruence D AAS Triangle Congruence E Not enough information

Slide 87 (Answer) / 183

40 What type of congruence exists, if any, between the two triangles? A SSS Triangle Congruence B SAS Triangle Congruence C ASA Triangle Congruence D AAS Triangle Congruence E Not enough information

[This object is a pull tab]

Answer D AAS Triangle Congruence Q: Are there vertical angles in this diagram?

Slide 88 / 183

A B C D 41 What type of congruence exists, if any, between the two triangles? A SSS Triangle Congruence B SAS Triangle Congruence C ASA Triangle Congruence D AAS Triangle Congruence E Not enough information

Given: BD bisects ∠ABC, ∠A ≅ ∠C

Slide 88 (Answer) / 183

A B C D 41 What type of congruence exists, if any, between the two triangles? A SSS Triangle Congruence B SAS Triangle Congruence C ASA Triangle Congruence D AAS Triangle Congruence E Not enough information

Given: BD bisects ∠ABC, ∠A ≅ ∠C

[This object is a pull tab]

Answer D Marking the bisected angle and the common side (reflexive) shows AAS.

Slide 89 / 183

42 What type of congruence exists, if any, between the two triangles? A SSS Triangle Congruence B SAS Triangle Congruence C ASA Triangle Congruence D AAS Triangle Congruence E Not enough information

Slide 89 (Answer) / 183

42 What type of congruence exists, if any, between the two triangles? A SSS Triangle Congruence B SAS Triangle Congruence C ASA Triangle Congruence D AAS Triangle Congruence E Not enough information

[This object is a pull tab]

Answer E Not Congruent There is no AAA Congruence. AAA does make them similar (same shape), but the size may be different.

Slide 90 / 183

HL Congruence

Return to Table

• f Contents

Slide 91 / 183

The final shortcut to proving congruence is Hypotenuse Leg Triangle Congruence, or the HL Triangle Congruence for short. This theorem states that if two right triangles have their hypotenuses and one of their legs congruent, then the triangles are congruent. The HL Triangle Congruence can be considered a corollary of the SSS Triangle Congruence.

Hypotenuse-Leg Triangle Congruence Slide 92 / 183

In a right triangle, the sum of the squares of the lengths of the two legs must equal the square of the length of the hypotenuse. c2 = a2 + b2 If we are given that for two right triangles the hypotenuse and one of the legs are equal (c1 = c2 and a1 = a2), then we know that the other leg but also be equal (b1 = b2). Thus, HL Triangle Congruence can be considered a special case, or corollary, of the Side-Side-Side Triangle Congruence.

Hypotenuse-Leg Triangle Congruence Slide 93 / 183

c12 = a12 + b12 c22 = a22 + b22 Solving for b in both equations c12 - a12 = b12 c22 - a22 = b22 b12 = c12 - a12 b22 = c22 - a22 Substituting c1 = c2 and a1 = a2 b12 = c22 - a22 b22 = c22 - a22 b12 = b22 b1 = b2

Hypotenuse-Leg Triangle Congruence Slide 93 (Answer) / 183

c12 = a12 + b12 c22 = a22 + b22 Solving for b in both equations c12 - a12 = b12 c22 - a22 = b22 b12 = c12 - a12 b22 = c22 - a22 Substituting c1 = c2 and a1 = a2 b12 = c22 - a22 b22 = c22 - a22 b12 = b22 b1 = b2

Hypotenuse-Leg Triangle Congruence

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Math Practice

This algebraic proof addresses MP2 & MP3

Slide 94 / 183

A B C R S T

Example

Are these two triangles congruent? These are right triangles, so look for HL Triangle Congruence.

Slide 94 (Answer) / 183

A B C R S T

Example

Are these two triangles congruent? These are right triangles, so look for HL Triangle Congruence.

[This object is a pull tab]

Math Practice

This example (and the next slide) addresses MP3 & MP6 Additional Q's to address MP Standards: What information are we given? (MP1) What do we need to prove? (MP1) What properties/theorems/postulates might help you? (MP1) How are the triangles related? Why? (MP6 & MP3)

Slide 95 / 183 Example

Recall that the side opposite the right angle is the hypotenuse, and the other two sides are called legs. Hypotenuse: AC ≅ RT Leg: CB ≅ TS By the HL Triangle Congruence, ∆ABC ≅ ∆RST A B C R S T

Slide 96 / 183

Side-Side-Side (SSS): three sides Side-Angle-Side (SAS): two sides and the included angle Angle-Side-Angle (ASA): two angles and the included side Angle-Angle-Side (AAS): two angles and one non-included side Hypotenuse-Leg (HL): hypotenuse and one leg (right triangles)

Postulates/Theorems to Prove Triangles Congruent

To use the congruence postulates/theorems, we need to know

• r be able to show the following congruences between two

triangles:

Slide 97 / 183

Given: Q R S X Y Z Mark the given

• n the diagram.

Note that it is a right triangle. 43 What type of congruence exists, if any, between the two triangles? A SSS Triangle Congruence B SAS Triangle Congruence C ASA Triangle Congruence D AAS Triangle Congruence E HL Triangle Congruence F Not enough information Hint

Click to reveal

Slide 97 (Answer) / 183

Given: Q R S X Y Z Mark the given

• n the diagram.

Note that it is a right triangle. 43 What type of congruence exists, if any, between the two triangles? A SSS Triangle Congruence B SAS Triangle Congruence C ASA Triangle Congruence D AAS Triangle Congruence E HL Triangle Congruence F Not enough information Hint

Click to reveal

[This object is a pull tab]

Answer

E

Q R S X Y Z

Slide 98 / 183

L M N O P Q If they are congruent what is the congruence statement? 44 What type of congruence exists, if any, between the two triangles? A SSS Triangle Congruence B SAS Triangle Congruence C ASA Triangle Congruence D AAS Triangle Congruence E HL Triangle Congruence F Not enough information

Slide 98 (Answer) / 183

L M N O P Q If they are congruent what is the congruence statement? 44 What type of congruence exists, if any, between the two triangles? A SSS Triangle Congruence B SAS Triangle Congruence C ASA Triangle Congruence D AAS Triangle Congruence E HL Triangle Congruence F Not enough information

[This object is a pull tab]

Answer

C Slide 99 / 183

A B C D E F If they are congruent what is the congruence statement? 45 What type of congruence exists, if any, between the two triangles? A SSS Triangle Congruence B SAS Triangle Congruence C ASA Triangle Congruence D AAS Triangle Congruence E HL Triangle Congruence F Not enough information

Slide 99 (Answer) / 183

A B C D E F If they are congruent what is the congruence statement? 45 What type of congruence exists, if any, between the two triangles? A SSS Triangle Congruence B SAS Triangle Congruence C ASA Triangle Congruence D AAS Triangle Congruence E HL Triangle Congruence F Not enough information

[This object is a pull tab]

Answer

F Slide 100 / 183

T U V W X Y If they are congruent what is the congruence statement? 46 What type of congruence exists, if any, between the two triangles? A SSS Triangle Congruence B SAS Triangle Congruence C ASA Triangle Congruence D AAS Triangle Congruence E HL Triangle Congruence F Not enough information

Slide 100 (Answer) / 183

T U V W X Y If they are congruent what is the congruence statement? 46 What type of congruence exists, if any, between the two triangles? A SSS Triangle Congruence B SAS Triangle Congruence C ASA Triangle Congruence D AAS Triangle Congruence E HL Triangle Congruence F Not enough information

[This object is a pull tab]

Answer

B

Slide 101 / 183

Q W E Y If they are congruent what is the congruence statement? 47 What type of congruence exists, if any, between the two triangles? A SSS Triangle Congruence B SAS Triangle Congruence C ASA Triangle Congruence D AAS Triangle Congruence E HL Triangle Congruence F Not enough information

Slide 101 (Answer) / 183

Q W E Y If they are congruent what is the congruence statement? 47 What type of congruence exists, if any, between the two triangles? A SSS Triangle Congruence B SAS Triangle Congruence C ASA Triangle Congruence D AAS Triangle Congruence E HL Triangle Congruence F Not enough information

[This object is a pull tab]

Answer

D Slide 102 / 183

N M O J K L If they are congruent what is the congruence statement? 48 What type of congruence exists, if any, between the two triangles? A SSS Triangle Congruence B SAS Triangle Congruence C ASA Triangle Congruence D AAS Triangle Congruence E HL Triangle Congruence F Not enough information

Slide 102 (Answer) / 183

N M O J K L If they are congruent what is the congruence statement? 48 What type of congruence exists, if any, between the two triangles? A SSS Triangle Congruence B SAS Triangle Congruence C ASA Triangle Congruence D AAS Triangle Congruence E HL Triangle Congruence F Not enough information

[This object is a pull tab]

Answer

E Slide 103 / 183

E F G H If they are congruent what is the congruence statement? 49 What type of congruence exists, if any, between the two triangles? A SSS Triangle Congruence B SAS Triangle Congruence C ASA Triangle Congruence D AAS Triangle Congruence E HL Triangle Congruence F Not enough information

Slide 103 (Answer) / 183

E F G H If they are congruent what is the congruence statement? 49 What type of congruence exists, if any, between the two triangles? A SSS Triangle Congruence B SAS Triangle Congruence C ASA Triangle Congruence D AAS Triangle Congruence E HL Triangle Congruence F Not enough information

[This object is a pull tab]

Answer

E

Slide 104 / 183

If they are congruent what is the congruence statement? 50 What type of congruence exists, if any, between the two triangles? A SSS Triangle Congruence B SAS Triangle Congruence C ASA Triangle Congruence D AAS Triangle Congruence E HL Triangle Congruence F Not enough information E F G H

Slide 104 (Answer) / 183

If they are congruent what is the congruence statement? 50 What type of congruence exists, if any, between the two triangles? A SSS Triangle Congruence B SAS Triangle Congruence C ASA Triangle Congruence D AAS Triangle Congruence E HL Triangle Congruence F Not enough information E F G H

[This object is a pull tab]

Answer

B Slide 105 / 183

If they are congruent what is the congruence statement? 51 What type of congruence exists, if any, between the two triangles? A SSS Triangle Congruence B SAS Triangle Congruence C ASA Triangle Congruence D AAS Triangle Congruence E HL Triangle Congruence F Not enough information K F B M

Slide 105 (Answer) / 183

If they are congruent what is the congruence statement? 51 What type of congruence exists, if any, between the two triangles? A SSS Triangle Congruence B SAS Triangle Congruence C ASA Triangle Congruence D AAS Triangle Congruence E HL Triangle Congruence F Not enough information K F B M

[This object is a pull tab]

Answer

A Slide 106 / 183

P O Y alternate interior If they are congruent what is the congruence statement? U What angles are congruent when parallel lines are cut by a transversal? 52 What type of congruence exists, if any, between the two triangles? A SSS Triangle Congruence B SAS Triangle Congruence C ASA Triangle Congruence D AAS Triangle Congruence E HL Triangle Congruence F Not enough information

Click to Reveal

Slide 106 (Answer) / 183

P O Y alternate interior If they are congruent what is the congruence statement? U What angles are congruent when parallel lines are cut by a transversal? 52 What type of congruence exists, if any, between the two triangles? A SSS Triangle Congruence B SAS Triangle Congruence C ASA Triangle Congruence D AAS Triangle Congruence E HL Triangle Congruence F Not enough information

Click to Reveal

[This object is a pull tab]

Answer

D

Slide 107 / 183

O K M J If they are congruent what is the congruence statement? 53 What type of congruence exists, if any, between the two triangles? A SSS Triangle Congruence B SAS Triangle Congruence C ASA Triangle Congruence D AAS Triangle Congruence E HL Triangle Congruence F Not enough information

Slide 107 (Answer) / 183

O K M J If they are congruent what is the congruence statement? 53 What type of congruence exists, if any, between the two triangles? A SSS Triangle Congruence B SAS Triangle Congruence C ASA Triangle Congruence D AAS Triangle Congruence E HL Triangle Congruence F Not enough information

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Answer

F Slide 108 / 183

A S X Z If they are congruent what is the congruence statement? 54 What type of congruence exists, if any, between the two triangles? A SSS Triangle Congruence B SAS Triangle Congruence C ASA Triangle Congruence D AAS Triangle Congruence E HL Triangle Congruence F Not enough information

Slide 108 (Answer) / 183

A S X Z If they are congruent what is the congruence statement? 54 What type of congruence exists, if any, between the two triangles? A SSS Triangle Congruence B SAS Triangle Congruence C ASA Triangle Congruence D AAS Triangle Congruence E HL Triangle Congruence F Not enough information

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Answer

E Slide 109 / 183

Triangle Congruence Proofs

Return to Table

• f Contents

Slide 110 / 183

First: identify given information Second: use a diagram that is marked with given information Third: review congruence postulates/theorems - what information is needed (sides/angles) to use

• ne of these?

SSS SAS ASA AAS HL

Strategy to Prove Congruence

Slide 110 (Answer) / 183

First: identify given information Second: use a diagram that is marked with given information Third: review congruence postulates/theorems - what information is needed (sides/angles) to use

• ne of these?

SSS SAS ASA AAS HL

Strategy to Prove Congruence

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Math Practice

This example (this slide and the next two) addresses MP1, MP3 & MP6

Slide 111 / 183

E F M G 90° 90° 8 8 H

Example

Given: MF = MH = 8 and m∠F = m∠H = 90º Prove: ΔEFM ≅ ΔGHM

Slide 112 / 183

55 Provide the reason for line 3.

A Angle-Side-Angle Triangle Congruence B Side-Side-Side Triangle Congruence C Side-Angle-Side Triangle Congruence D Vertical angles are congruent E Alternate interior angles are congruent

E F M G 90° 90° 8 8 H

Given: MF = MH = 8 and m∠F = m∠H = 90º Prove: ΔEFM ≅ ΔGHM

Statement Reason 1 MF = MH = 8 and m∠F = m∠H = 90º Given 2 MF ≅ MH and ∠F ≅ ∠H

• Defn. of

congruence 3 ∠FME ≅ ∠HMG ? 4 ΔEFM ≅ ΔGHM ?

Answer

Slide 113 / 183

56 Provide the reason for line 4.

A Angle-Side-Angle Triangle Congruence B Side-Side-Side Triangle Congruence C Side-Angle-Side Triangle Congruence D Vertical angles are congruent E Alternate interior angles are congruent

E F M G 90° 90° 8 8 H

Given: MF = MH = 8 and m∠F = m∠H = 90º Prove: ΔEFM ≅ ΔGHM

Statement Reason 1 MF = MH = 8 and m∠F = m∠H = 90º Given 2 MF ≅ MH and ∠F ≅ ∠H

• Defn. of

congruence 3 ∠FME ≅ ∠HMG ? 4 ΔEFM ≅ ΔGHM ?

Answer

Slide 114 / 183 Congruent Reasons Summary SSS SSA SAS AAS ASA AAA HL 3 1 2

# of congruent angles

postulate/ theorem (Drag those that don't work out of the chart. Then put HL where it would belong.)

Slide 114 (Answer) / 183 Congruent Reasons Summary SSS SSA SAS AAS ASA AAA HL 3 1 2

# of congruent angles

postulate/ theorem (Drag those that don't work out of the chart. Then put HL where it would belong.)

[This object is a pull tab]

Answer SSS SAS AAS ASA 3 1 2

# of congruent angles postulate/ theorem

SSA AAA HL

Slide 115 / 183 Example

A F K G B Solution (two-column): 1) Given 2) SSS Triangle Congruence AF ≅ BG, FK ≅ GK KA ≅ KB 1) 2) ΔAFK ≅ ΔBGK Statements Reasons Given: AF ≅ BG, FK ≅ GK, and KA ≅ KB Prove: ΔAFK ≅ ΔBGK

Slide 115 (Answer) / 183 Example

A F K G B Solution (two-column): 1) Given 2) SSS Triangle Congruence AF ≅ BG, FK ≅ GK KA ≅ KB 1) 2) ΔAFK ≅ ΔBGK Statements Reasons Given: AF ≅ BG, FK ≅ GK, and KA ≅ KB Prove: ΔAFK ≅ ΔBGK

[This object is a pull tab]

Math Practice

This example addresses MP3 & MP6

Slide 116 / 183

• 3. Reflexive property
• 2. Definition of ∠bisector
• 4. SAS Triangle Congruence

Example

A B C D

Statements Reasons

• 1. Given

click ___________ click ___________ click ___________

Given: BC ≅ CD AC bisects ∠BCD Prove: ∆ABC ≅ ∆ADC

• 1. BC ≅ CD, AC bisects ∠BCD
• 2. ∠BCA ≅ ∠DCA
• 3. AC ≅ AC
• 4. ∆ABC ≅ ∆ADC

click

Slide 116 (Answer) / 183

• 3. Reflexive property
• 2. Definition of ∠bisector
• 4. SAS Triangle Congruence

Example

A B C D

Statements Reasons

• 1. Given

click ___________ click ___________ click ___________

Given: BC ≅ CD AC bisects ∠BCD Prove: ∆ABC ≅ ∆ADC

• 1. BC ≅ CD, AC bisects ∠BCD
• 2. ∠BCA ≅ ∠DCA
• 3. AC ≅ AC
• 4. ∆ABC ≅ ∆ADC

click [This object is a pull tab]

Math Practice

This example (this slide and the next) addresses MP3 & MP6 Additional Q's to address MP Standards: What information are we given? (MP1) What do we need to prove? (MP1) What properties/theorems/postulates might help you? (MP1) How are the triangles related? Why? (MP6 & MP3)

Slide 117 / 183 Problem

D F G E

Write a two-column proof. Given: DE ‖ FG DE ≅ FG Prove: ∆DEG ≅ ∆FGE

Slide 117 (Answer) / 183 Problem

D F G E

Write a two-column proof. Given: DE ‖ FG DE ≅ FG Prove: ∆DEG ≅ ∆FGE

[This object is a pull tab]

Math Practice

This example (this slide and the next) addresses MP3 & MP6 Additional Q's to address MP Standards: What information are we given? (MP1) What do we need to prove? (MP1) What properties/theorems/postulates might help you? (MP1) How could you use the drawing to show your thinking? (MP5) How are the triangles related? Why? (MP6 & MP3)

Slide 118 / 183 Problem

D F G E Statements Reasons

• 1. DE ‖ FG
• 2. ∠DEG ≅ ∠FGE

DE ≅ FG

• 3. GE ≅ EG
• 4. ΔDEG ≅ ΔFGE
• 1. Given
• 2. Alternate Interior Angles are

≅

• 3. Reflexive Property of Congruence
• 4. SAS Triangle Congruence

Click Click Click Click

Given: DE ‖ FG DE ≅ FG

Slide 119 / 183

A C D T B

Problem: complete the proof

___________

Given: ∠A and ∠D are right angles; AT ≅ DT Prove: ΔATB ≅ ΔDTC

• 1. ∠A and ∠D are right angles
• 2. ∠A ≅ ∠D
• 3. AT ≅ DT
• 4. ∠ ATB ≅ ∠ DTC
• 5. ΔATB ≅ ΔDTC

Click Click Click

• 1. Given
• 2. right ∠'s are congruent
• 3. Given
• 4. Vertical ∠'s are congruent
• 5. ASA Triangle Congruence

Click Click Click

Statements Reasons

Click

Slide 119 (Answer) / 183

A C D T B

Problem: complete the proof

___________

Given: ∠A and ∠D are right angles; AT ≅ DT Prove: ΔATB ≅ ΔDTC

• 1. ∠A and ∠D are right angles
• 2. ∠A ≅ ∠D
• 3. AT ≅ DT
• 4. ∠ ATB ≅ ∠ DTC
• 5. ΔATB ≅ ΔDTC

Click Click Click

• 1. Given
• 2. right ∠'s are congruent
• 3. Given
• 4. Vertical ∠'s are congruent
• 5. ASA Triangle Congruence

Click Click Click

Statements Reasons

Click

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Math Practice

This example addresses MP3 & MP6 Additional Q's to address MP Standards: What information are we given? (MP1) What do we need to prove? (MP1) What properties/theorems/postulates might help you? (MP1) How could you use the drawing to show your thinking? (MP5) How are the triangles related? Why? (MP6 & MP3)

Slide 120 / 183 Problem: complete the proof

D C A B Given: DA ⊥ AB Prove: ΔDAB ≅ ΔBCD BC ⊥ CD ∠ADB ≅ ∠CBD Statements Reasons

• 1. DA ⊥ AB, BC ⊥ CD
• 2. ∠A and ∠C are right ∠'s
• 3. ∠A ≅ ∠C
• 4. ∠ADB ≅ ∠CBD
• 5. DB ≅ BD
• 6. ΔDAB ≅ ΔBCD
• 1. Given
• 2. Definition of ⊥ lines
• 3. All right angles are congruent
• 4. Given
• 5. reflexive property of ≅
• 6. AAS Triangle Congruence

Click Click Click Click Click Click Click Click Click

Slide 120 (Answer) / 183 Problem: complete the proof

D C A B Given: DA ⊥ AB Prove: ΔDAB ≅ ΔBCD BC ⊥ CD ∠ADB ≅ ∠CBD Statements Reasons

• 1. DA ⊥ AB, BC ⊥ CD
• 2. ∠A and ∠C are right ∠'s
• 3. ∠A ≅ ∠C
• 4. ∠ADB ≅ ∠CBD
• 5. DB ≅ BD
• 6. ΔDAB ≅ ΔBCD
• 1. Given
• 2. Definition of ⊥ lines
• 3. All right angles are congruent
• 4. Given
• 5. reflexive property of ≅
• 6. AAS Triangle Congruence

Click Click Click Click Click Click Click Click Click

[This object is a pull tab]

Math Practice

This example addresses MP3 & MP6 Additional Q's to address MP Standards: What information are we given? (MP1) What do we need to prove? (MP1) What properties/theorems/postulates might help you? (MP1) How could you use the drawing to show your thinking? (MP5) How are the triangles related? Why? (MP6 & MP3)

Slide 121 / 183

Statements Reason s 1) 2) 3) 4) 5) 1) 2) 3) 4) 5) Given: AC ≅ BD, E is the midpoint of AB and CD Prove: ΔAEC ≅ ΔBED A B D C E

Problem

E is the midpoint of AB and CD SSS Triangle Congruence AC = BD ~

• Def. of midpoint

AE = BE ~ Given ∆AEC ≅ ∆BED CE = DE ~

Slide 121 (Answer) / 183

Statements Reason s 1) 2) 3) 4) 5) 1) 2) 3) 4) 5) Given: AC ≅ BD, E is the midpoint of AB and CD Prove: ΔAEC ≅ ΔBED A B D C E

Problem

E is the midpoint of AB and CD SSS Triangle Congruence AC = BD ~

• Def. of midpoint

AE = BE ~ Given ∆AEC ≅ ∆BED CE = DE ~

Teacher Notes

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Given 1) 2) 3) 4) 5) Statements Reasons 1) 2) 3) 4) 5) SSS Triangle Congruence

• Def. of midpoint
• Def. of midpoint

AC = BD ~ CE = DE ~ AE = BE ~ ~ AEC = BED E is the midpoint of AB and CD Given

This example can be solved by matching the statements & reasons with their appropriate location.

• Ans. is given below.

Slide 122 / 183

Return to Table

• f Contents

CPCTC

Corresponding Parts of Congruent Triangles are Congruent

Slide 123 / 183

CPCTC states that if two or more triangles are congruent by

• ne of the congruence postulates/theorems -

SSS, SAS, ASA, AAS, or HL, then all of their corresponding parts are also congruent. Corresponding Parts of Congruent Triangles are Congruent

CPCTC

Sometimes, our goal is not to prove two triangles congruent, but to show that a pair of corresponding sides or angles are congruent, or that some

• ther property is true.

Slide 124 / 183 Process for proving that two segments

• r angles are congruent
• 1. Find two triangles in which the two sides
• r two angles are corresponding parts
• 2. Prove that the two triangles are congruent

(SSS, SAS, ASA, AAS, HL)

• 3. State that the two parts are congruent, using as the reason:

"Corresponding Parts of Congruent Triangles are Congruent"

Slide 125 / 183

M N O E L 57 Which two triangles might you try to prove congruent in order to prove NM ≅ NO ? A B C D ΔLOE ΔNOL ΔLME ΔNME

Slide 125 (Answer) / 183

M N O E L 57 Which two triangles might you try to prove congruent in order to prove NM ≅ NO ? A B C D ΔLOE ΔNOL ΔLME ΔNME

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Answer

B and D

Slide 126 / 183

58 Which two triangles might you try to prove congruent in order to prove EO ≅ LM ? A B C D ΔEOL ΔNOL ΔLME ΔNME M N O E L

Slide 126 (Answer) / 183

58 Which two triangles might you try to prove congruent in order to prove EO ≅ LM ? A B C D ΔEOL ΔNOL ΔLME ΔNME M N O E L

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Answer

A and C Slide 127 / 183

59 Which two triangles might you try to prove congruent in order to prove ∠1 ≅ ∠2? A B C D ΔLOE ΔNOL ΔLME ΔNME M N O E L 1 2 Answer

Slide 128 / 183

60 Which two triangles might you try to prove congruent in order to prove EN ≅ LN ? A B C D ΔLOE ΔNOL ΔLME ΔNME M N O E L

Slide 128 (Answer) / 183

60 Which two triangles might you try to prove congruent in order to prove EN ≅ LN ? A B C D ΔLOE ΔNOL ΔLME ΔNME M N O E L

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Answer

B and D Slide 129 / 183

Statements Reasons

Problem: complete the proof

Given: AB ≅ DE, BC ≅ EC, C is the midpoint of AD Prove: ∠A ≅ ∠D A B C D E

• 1. AB ≅ DE
• 2. BC ≅ EC
• 3. C is the midpoint of AD
• 4. CA ≅ CD
• 5. ΔABC ≅ ΔDEC
• 6. ∠A ≅ ∠D
• 1. Given
• 2. Given
• 3. Given
• 4. Definition of midpoint
• 5. SSS Triangle Congruence
• 6. CPCTC

Click Click Click Click Click Click Click

Slide 129 (Answer) / 183

Statements Reasons

Problem: complete the proof

Given: AB ≅ DE, BC ≅ EC, C is the midpoint of AD Prove: ∠A ≅ ∠D A B C D E

• 1. AB ≅ DE
• 2. BC ≅ EC
• 3. C is the midpoint of AD
• 4. CA ≅ CD
• 5. ΔABC ≅ ΔDEC
• 6. ∠A ≅ ∠D
• 1. Given
• 2. Given
• 3. Given
• 4. Definition of midpoint
• 5. SSS Triangle Congruence
• 6. CPCTC

Click Click Click Click Click Click Click

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Math Practice

This example addresses MP3 & MP6 Additional Q's to address MP Standards: What information are we given? (MP1) What do we need to prove? (MP1) What properties/theorems/postulates might help you? (MP1) How could you use the drawing to show your thinking? (MP5) How are the triangles related? Why? (MP6 & MP3)

Slide 130 / 183 Problem

A B C D E

We are given that ∠BCA ≅ ∠DCE, BC ≅ CD, and ∠B and ∠D are right angles. Since all right angles are congruent, ∠B ≅ ∠D. With the congruent angles and segments, we can conclude that ΔABC ≅ ΔEDC by ASA. Therefore, BA ≅ DE by CPCTC. Given: ∠BCA ≅ ∠DCE ∠B and ∠D are right angles BC ≅ CD Prove: BA ≅ DE

Click Click Click Click Click

Slide 130 (Answer) / 183 Problem

A B C D E

We are given that ∠BCA ≅ ∠DCE, BC ≅ CD, and ∠B and ∠D are right angles. Since all right angles are congruent, ∠B ≅ ∠D. With the congruent angles and segments, we can conclude that ΔABC ≅ ΔEDC by ASA. Therefore, BA ≅ DE by CPCTC. Given: ∠BCA ≅ ∠DCE ∠B and ∠D are right angles BC ≅ CD Prove: BA ≅ DE

Click Click Click Click Click [This object is a pull tab]

Math Practice

This example addresses MP3 & MP6 Additional Q's to address MP Standards: What information are we given? (MP1) What do we need to prove? (MP1) What properties/theorems/postulates might help you? (MP1) How could you use the drawing to show your thinking? (MP5) How are the triangles related? Why? (MP6 & MP3)

Slide 131 / 183

Statements Reasons

Problem: complete the proof

W X P Z Y

Given: P is the midpoint of WY,

P is the midpoint of XZ

Prove: WX ‖ ZY

• 1. P is the midpoint of WY
• 2. P is the midpoint of XZ
• 3. WP ≅ YP, ZP ≅ XP
• 4. ∠WPX ≅ ∠YPZ
• 5. ΔWPX ≅ ΔYPZ
• 6. ∠Z ≅ ∠X
• 7. WX ‖ ZY
• 1. Given
• 2. Given
• 3. Definition of midpoint
• 4. Vertical angles are congruent
• 5. SAS Triangle Congruence
• 6. CPCTC
• 7. If alt. int. angles are congruent,

then lines are parallel

Click

Click Click Click

Click Click Click Click Click Click

Slide 131 (Answer) / 183

Statements Reasons

Problem: complete the proof

W X P Z Y

Given: P is the midpoint of WY,

P is the midpoint of XZ

Prove: WX ‖ ZY

• 1. P is the midpoint of WY
• 2. P is the midpoint of XZ
• 3. WP ≅ YP, ZP ≅ XP
• 4. ∠WPX ≅ ∠YPZ
• 5. ΔWPX ≅ ΔYPZ
• 6. ∠Z ≅ ∠X
• 7. WX ‖ ZY
• 1. Given
• 2. Given
• 3. Definition of midpoint
• 4. Vertical angles are congruent
• 5. SAS Triangle Congruence
• 6. CPCTC
• 7. If alt. int. angles are congruent,

then lines are parallel

Click

Click Click Click

Click Click Click Click Click Click

[This object is a pull tab]

Math Practice

This example addresses MP3 & MP6 Additional Q's to address MP Standards: What information are we given? (MP1) What do we need to prove? (MP1) What properties/theorems/postulates might help you? (MP1) How could you use the drawing to show your thinking? (MP5) How are the triangles related? Why? (MP6 & MP3)

Slide 132 / 183 Additional Proof Practice

Website link: Interactive Proofs

Slide 132 (Answer) / 183 Additional Proof Practice

Website link: Interactive Proofs

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Teacher Notes

URL for the website: http:// feromax.com/cgi-bin/ProveIt.pl Proofs for Congruent Triangles can be found at the bottom of the

• list. All require CPCTC.

They could be done as a class, or in small groups w/ the use of laptops, iPads, Chromebooks, etc.

Slide 133 / 183

Learning mathematics is like climbing a ladder, one step leads to the next. No step is more difficult than the one before it, as long as you take them one step at a time. Congruent Triangles are an important step in geometry. They will be used through much of the rest of this course. For example, the following PARCC-type question looks like it's about parallelograms, but you can answer every part of this question with what you know already, before you even study quadrilaterals. Try it out.

Using What You've Learned Slide 134 / 183

A B C D

Given: ∠BAC ≅ ∠DCA, BA ≅ DC Prove: ABCD is a parallelogram Statements Reasons

• 1. ∠BAC ≅ ∠DCA, BA ≅ DC
• 2. AC ≅ CA
• 3. ΔBAC ≅ ΔDCA
• 4. ∠BCA ≅ ∠DAC
• 5. BC || AD, AB || DC
• 6. ABCD is a parallelogram
• 1. Given
• 2. Reflexive Property of Congruence
• 3. SAS Triangle Congruence
• 4. CPCTC
• 5. If alt. int. angles are congruent,

then lines are parallel

• 6. Definition of a parallelogram

Click

Click

Click Click Click Click

Slide 135 / 183

Isosceles Triangle Theorems

Return to Table

• f Contents

Slide 136 / 183 Isosceles Triangles

leg leg base In an isosceles triangle, the base is the side that is not necessarily congruent to the

• ther two sides (legs).

If an isosceles triangle has 3 congruent sides, it is also an equilateral triangle.

Slide 137 / 183

base angles vertex angle

Isosceles Triangles

The vertex angle is

• pposite the base, and is

the included angle between the legs. The base angles are the angles opposite the legs, and are included by a leg and the base.

Slide 138 / 183 Base Angles Theorem

The base angles of an isosceles triangle are congruent. This says that the angles opposite equal sides of a triangle are of equal measure. In isosceles triangles the angles at the base are equal to one another, and, if the equal straight lines be produced further, the angles under the base will be equal to one another. Euclid: Book One Proposition 5

Slide 139 / 183 Proof of Base Angles Theorem

Given: In ΔABC, AB ≅ BC Prove: ∠A ≅ ∠C

A B C

There are several ways to prove this. Euclid's way is pretty complicated. The link below shows two typical proofs and an alternate third one. The third proof uses the fact that

• rder DOES matter in making

statements of congruence. It was supposedly generated by a computer.

http://www.qc.edu.hk/math/Junior%20Secondary/isosceles%20triangle.htm

Slide 139 (Answer) / 183 Proof of Base Angles Theorem

Given: In ΔABC, AB ≅ BC Prove: ∠A ≅ ∠C

A B C

There are several ways to prove this. Euclid's way is pretty complicated. The link below shows two typical proofs and an alternate third one. The third proof uses the fact that

• rder DOES matter in making

statements of congruence. It was supposedly generated by a computer.

http://www.qc.edu.hk/math/Junior%20Secondary/isosceles%20triangle.htm

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Math Practice

This example addresses MP3 & MP6 Additional Q's to address MP Standards: What information are we given? (MP1) What do we need to prove? (MP1) What properties/theorems/postulates might help you? (MP1) How could you use the drawing to show your thinking? (MP5) How are the triangles related? Why? (MP6 & MP3)

Slide 140 / 183

Statement Reason 1 In ΔABC, AB ≅ BC Given 2 BC ≅ AB Symmetric Property of ≅ 3 ∠ABC ≅ ∠CBA Reflexive Property of ≅ 4 ΔABC ≅ ΔCBA SAS Triangle Congruence 5 ∠A ≅ ∠C CPCTC

Proof of Base Angles Theorem

Given: In ΔABC, AB ≅ BC Prove: ∠A ≅ ∠C

A B C

Below are the arguments that could be used to explain the third proof from the link on the previous slide (computer generated).

Slide 141 / 183

61 What is the value of x in this triangle? Justify your answer. x° y° 44°

Slide 141 (Answer) / 183

61 What is the value of x in this triangle? Justify your answer. x° y° 44°

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Answer

44º

Slide 142 / 183

62 What is the value of y in this triangle? Justify your answer. x° y° 44°

Slide 142 (Answer) / 183

62 What is the value of y in this triangle? Justify your answer. x° y° 44°

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Answer

92º

Slide 143 / 183

x° y° 72° 63 Solve for x and y. Explain your reasoning.

Slide 143 (Answer) / 183

x° y° 72° 63 Solve for x and y. Explain your reasoning.

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Answer x = 36 y = 72

Slide 144 / 183

64 What is the measure of each base angle? 70°

Slide 144 (Answer) / 183

64 What is the measure of each base angle? 70°

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Answer 70 + 2x = 180 2x = 110 x = 55 Each base angle is 55º

70° x° x°

Slide 145 / 183

65 The vertex angle of an isosceles triangle is 38°. What is the measure of each base angle? A 71° B 38° C 83° D 104°

Slide 145 (Answer) / 183

65 The vertex angle of an isosceles triangle is 38°. What is the measure of each base angle? A 71° B 38° C 83° D 104°

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Answer

A Slide 146 / 183 Converse of Base Angles Theorem

If two angles of a triangle are congruent, then the sides opposite them are congruent. The sides opposite equal angles of a triangle are of equal length. If in a triangle two angles be equal to one another, the sides opposite those angles will also be equal to one another Euclid: Book One Proposition 6

Slide 147 / 183

D E F

4 3

66 What is the length of FD?

Slide 147 (Answer) / 183

D E F

4 3

66 What is the length of FD?

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Answer

FD = 4

Slide 148 / 183

D E F

9 7

67 What is the length of ED?

Slide 148 (Answer) / 183

D E F

9 7

67 What is the length of ED?

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Answer

ED = 9

Slide 149 / 183 Equilateral Triangles

If three sides of a triangle are equal, each of the three angles has a measure of 60º . An equilateral triangle is a special case of an isosceles triangle. The base and legs are all of equal length. Since all the angles are opposite sides of equal length, they all have equal measure. Since the three angles add to 180º and have equal measure, they each have a measure of 60º. Conversely, if two angles of a triangle are each 60º, the third angle also has a measure of 60º and all the sides are of equal length.

Slide 150 / 183 Equilateral Triangles

If three sides of a triangle are equal, each of the three angles has a measure of 60º . Conversely, if the angles of a triangle each have a measure of 60º, all the sides are of equal length. Also, if two angles of a triangle each have a measure of 60º, the third angle must also has a measure of 60º since the Interior Angles Theorem indicates that the angles must add to 180º Then, all the sides must be of equal length.

Slide 151 / 183

68 Classify the triangle by sides and angles. A equilateral B isosceles C scalene D equiangular E acute F

• btuse

G right

7 40º

Slide 151 (Answer) / 183

68 Classify the triangle by sides and angles. A equilateral B isosceles C scalene D equiangular E acute F

• btuse

G right

7 40º

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Answer

B and E Slide 152 / 183

69 Classify the triangle by sides and angles. A equilateral B isosceles C scalene D equiangular E acute F

• btuse

G right

4 4 4

Slide 152 (Answer) / 183

69 Classify the triangle by sides and angles. A equilateral B isosceles C scalene D equiangular E acute F

• btuse

G right

4 4 4

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Answer

A, B, D & E Slide 153 / 183

5 113º 3 3

70 Classify the triangle by sides and angles. A equilateral B isosceles C scalene D equiangular E acute F

• btuse

G right

Slide 153 (Answer) / 183

5 113º 3 3

70 Classify the triangle by sides and angles. A equilateral B isosceles C scalene D equiangular E acute F

• btuse

G right

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Answer

B and F Slide 154 / 183

71 Classify the triangle by sides and angles. A equilateral B isosceles C scalene D equiangular E acute F

• btuse

G right

12 12

Slide 154 (Answer) / 183

71 Classify the triangle by sides and angles. A equilateral B isosceles C scalene D equiangular E acute F

• btuse

G right

12 12

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Answer

A, B, D & E Slide 155 / 183

72 Classify the triangle by sides and angles. A equilateral B isosceles C scalene D equiangular E acute F

• btuse

G right

60º 60º

Slide 155 (Answer) / 183

72 Classify the triangle by sides and angles. A equilateral B isosceles C scalene D equiangular E acute F

• btuse

G right

60º 60º

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Answer

A, B, D & E Slide 156 / 183 Example

Find the value of x and y. Explain your reasoning.

y° x°

Slide 156 (Answer) / 183 Example

Find the value of x and y. Explain your reasoning.

y° x°

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Answer

x = 60º y = 30º

Additional Q's to address MP standards: What do you know about Triangle Sum Theorem & Base Angles Theorem that you can apply to this situation? (MP7) Can you find a shortcut to solve the problem? (MP8)

• Ans: the value of y will be half the

value of the angle that is adjacent to the

• btuse angle

Slide 157 / 183

73 What is the value of y? A 120° B 70° C 55° D 125° 70° y°

Slide 157 (Answer) / 183

73 What is the value of y? A 120° B 70° C 55° D 125° 70° y°

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Answer

D Slide 158 / 183

74 What is the value of x? Justify your answer. A 50° B 25° C 30° D 130° 50° x°

Slide 158 (Answer) / 183

74 What is the value of x? Justify your answer. A 50° B 25° C 30° D 130° 50° x°

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Answer

B Slide 159 / 183

3x - 17

28

75 Solve for x in the diagram. A 3 2/3 B 14 C 15 D 16

Slide 159 (Answer) / 183

3x - 17

28

75 Solve for x in the diagram. A 3 2/3 B 14 C 15 D 16

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Answer

C Slide 160 / 183 PARCC Sample Test Questions

The remaining slides in this presentation contain questions from the PARCC Sample Test. After finishing this unit, you should be able to answer these questions. Good Luck! Return to Table

• f Contents

Slide 161 / 183

76 The first step of the construction is to draw an arc centered at point A that intersects both sides of the given angle. What is established by the first step? A AB ≅ BC B AB ≅ AC C AD ≅ AC D BD ≅ CD

Use the information provided in the animation to answer the questions about the geometric

• construction. (note: an online video plays

demonstrating the construction)

A C B D Part A Question 18/25

Topic: Angle Constructions (Unit 2) & Triangle Congruence Proofs PARCC Released Question (EOY)

Slide 161 (Answer) / 183

76 The first step of the construction is to draw an arc centered at point A that intersects both sides of the given angle. What is established by the first step? A AB ≅ BC B AB ≅ AC C AD ≅ AC D BD ≅ CD

Use the information provided in the animation to answer the questions about the geometric

• construction. (note: an online video plays

demonstrating the construction)

A C B D Part A Question 18/25

Topic: Angle Constructions (Unit 2) & Triangle Congruence Proofs PARCC Released Question (EOY)

[This object is a pull tab]

Answer

B

Slide 162 / 183

77 The construction creates congruent triangles. ABD and ACD are congruent because of the _______________ postulate/theorem. A side-side-side B angle-side-angle C side-angle-side D angle-angle-side Use the information provided in the animation to answer the questions about the geometric

• construction. (note: an online video plays

demonstrating the construction)

A C B D

Part B Complete the sentence with the choices given below.

Question 18/25

Topic: Angle Constructions (Unit 2) & Triangle Congruence Proofs

Slide 162 (Answer) / 183

77 The construction creates congruent triangles. ABD and ACD are congruent because of the _______________ postulate/theorem. A side-side-side B angle-side-angle C side-angle-side D angle-angle-side Use the information provided in the animation to answer the questions about the geometric

• construction. (note: an online video plays

demonstrating the construction)

A C B D

Part B Complete the sentence with the choices given below.

Question 18/25

Topic: Angle Constructions (Unit 2) & Triangle Congruence Proofs

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Answer

A

A C B D

Slide 163 / 183

78 It follows that AD must be the angle bisector of ∠BAC because _________________. A ∠ACD ≅ ∠ABD B ∠BAC ≅ ∠BDC C ∠BAD ≅ ∠CAD D ∠BAD ≅ ∠ABD Use the information provided in the animation to answer the questions about the geometric

• construction. (note: an online video plays

demonstrating the construction)

A C B D

Part B Complete the sentence with the choices given below.

Question 18/25

Topic: Angle Constructions (Unit 2) & Triangle Congruence Proofs

Slide 163 (Answer) / 183

78 It follows that AD must be the angle bisector of ∠BAC because _________________. A ∠ACD ≅ ∠ABD B ∠BAC ≅ ∠BDC C ∠BAD ≅ ∠CAD D ∠BAD ≅ ∠ABD Use the information provided in the animation to answer the questions about the geometric

• construction. (note: an online video plays

demonstrating the construction)

A C B D

Part B Complete the sentence with the choices given below.

Question 18/25

Topic: Angle Constructions (Unit 2) & Triangle Congruence Proofs

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Answer

C

A C B D

Slide 164 / 183

Question 2/11

Topic: Angle Constructions (Unit 2) & Triangle Congruence Proofs Marcella drew each step of a construction of an angle bisector. Z B A Z Z B A Z B A C Z B A C

Step 1 Step 2 Step 3 Step 4 Step 5

Part A Angle Z is given in Step 1. Describe the instructions for Steps 2 through 5 of the construction. This is a great problem and draws on a lot of what we've learned. Try it in your groups.Then we'll work on it step by step together by asking questions that break the problem into pieces. PARCC Released Question (PBA)

Slide 164 (Answer) / 183

Question 2/11

Topic: Angle Constructions (Unit 2) & Triangle Congruence Proofs Marcella drew each step of a construction of an angle bisector. Z B A Z Z B A Z B A C Z B A C

Step 1 Step 2 Step 3 Step 4 Step 5

Part A Angle Z is given in Step 1. Describe the instructions for Steps 2 through 5 of the construction. This is a great problem and draws on a lot of what we've learned. Try it in your groups.Then we'll work on it step by step together by asking questions that break the problem into pieces. PARCC Released Question (PBA)

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Answer

Part A: Full Credit

Slide 165 / 183

79 Angle Z is given in Step 1. What would be the description used to get from Step 1 to Step 2? A Construct an arc located in the interior of angle Z using a compass centered at point B with a radius length that is congruent to the radius length used to draw the arc centered at point A. Label the intersection point of the 2 interior arcs point C. B Construct an arc located in the interior of angle Z using a compass centered at point A and a radius greater than half of angle ZBA. C Draw a ray ZC, which is the angle bisector of angle BZA. D Construct an arc using a compass centered at point Z and any radius

• length. Label the points where the arc intersects the angle A and B.

Question 2/11

Topic: Angle Constructions (Unit 2) & Triangle Congruence Proofs Marcella drew each step of a construction of an angle bisector. Z B A Z Z B A Z B A C Z B A C

Step 1 Step 2 Step 3 Step 4 Step 5

Slide 165 (Answer) / 183

79 Angle Z is given in Step 1. What would be the description used to get from Step 1 to Step 2? A Construct an arc located in the interior of angle Z using a compass centered at point B with a radius length that is congruent to the radius length used to draw the arc centered at point A. Label the intersection point of the 2 interior arcs point C. B Construct an arc located in the interior of angle Z using a compass centered at point A and a radius greater than half of angle ZBA. C Draw a ray ZC, which is the angle bisector of angle BZA. D Construct an arc using a compass centered at point Z and any radius

• length. Label the points where the arc intersects the angle A and B.

Question 2/11

Topic: Angle Constructions (Unit 2) & Triangle Congruence Proofs Marcella drew each step of a construction of an angle bisector. Z B A Z Z B A Z B A C Z B A C

Step 1 Step 2 Step 3 Step 4 Step 5

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Answer

D

Slide 166 / 183

80 What would be the description used to get from Step 2 to Step 3? A Construct an arc located in the interior of angle Z using a compass centered at point B with a radius length that is congruent to the radius length used to draw the arc centered at point A. Label the intersection point of the 2 interior arcs point C. B Construct an arc located in the interior of angle Z using a compass centered at point A and a radius greater than half of angle ZBA. C Draw a ray ZC, which is the angle bisector of angle BZA. D Construct an arc using a compass centered at point Z and any radius

• length. Label the points where the arc intersects the angle A and B.

Question 2/11

Topic: Angle Constructions (Unit 2) & Triangle Congruence Proofs Marcella drew each step of a construction of an angle bisector. Z B A Z Z B A Z B A C Z B A C

Step 1 Step 2 Step 3 Step 4 Step 5

Slide 166 (Answer) / 183

80 What would be the description used to get from Step 2 to Step 3? A Construct an arc located in the interior of angle Z using a compass centered at point B with a radius length that is congruent to the radius length used to draw the arc centered at point A. Label the intersection point of the 2 interior arcs point C. B Construct an arc located in the interior of angle Z using a compass centered at point A and a radius greater than half of angle ZBA. C Draw a ray ZC, which is the angle bisector of angle BZA. D Construct an arc using a compass centered at point Z and any radius

• length. Label the points where the arc intersects the angle A and B.

Question 2/11

Topic: Angle Constructions (Unit 2) & Triangle Congruence Proofs Marcella drew each step of a construction of an angle bisector. Z B A Z Z B A Z B A C Z B A C

Step 1 Step 2 Step 3 Step 4 Step 5

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Answer

B

Slide 167 / 183

81 What would be the description used to get from Step 3 to Step 4? A Construct an arc located in the interior of angle Z using a compass centered at point B with a radius length that is congruent to the radius length used to draw the arc centered at point A. Label the intersection point of the 2 interior arcs point C. B Construct an arc located in the interior of angle Z using a compass centered at point A and a radius greater than half of angle ZBA. C Draw a ray ZC, which is the angle bisector of angle BZA. D Construct an arc using a compass centered at point Z and any radius

• length. Label the points where the arc intersects the angle A and B.

Question 2/11

Topic: Angle Constructions (Unit 2) & Triangle Congruence Proofs Marcella drew each step of a construction of an angle bisector. Z B A Z Z B A Z B A C Z B A C

Step 1 Step 2 Step 3 Step 4 Step 5

Slide 167 (Answer) / 183

81 What would be the description used to get from Step 3 to Step 4? A Construct an arc located in the interior of angle Z using a compass centered at point B with a radius length that is congruent to the radius length used to draw the arc centered at point A. Label the intersection point of the 2 interior arcs point C. B Construct an arc located in the interior of angle Z using a compass centered at point A and a radius greater than half of angle ZBA. C Draw a ray ZC, which is the angle bisector of angle BZA. D Construct an arc using a compass centered at point Z and any radius

• length. Label the points where the arc intersects the angle A and B.

Question 2/11

Topic: Angle Constructions (Unit 2) & Triangle Congruence Proofs Marcella drew each step of a construction of an angle bisector. Z B A Z Z B A Z B A C Z B A C

Step 1 Step 2 Step 3 Step 4 Step 5

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Answer

A

Slide 168 / 183

82 What would be the description used to get from Step 4 to Step 5? A Construct an arc located in the interior of angle Z using a compass centered at point B with a radius length that is congruent to the radius length used to draw the arc centered at point A. Label the intersection point of the 2 interior arcs point C. B Construct an arc located in the interior of angle Z using a compass centered at point A and a radius greater than half of angle ZBA. C Draw a ray ZC, which is the angle bisector of angle BZA. D Construct an arc using a compass centered at point Z and any radius

• length. Label the points where the arc intersects the angle A and B.

Question 2/11

Topic: Angle Constructions (Unit 2) & Triangle Congruence Proofs Marcella drew each step of a construction of an angle bisector. Z B A Z Z B A Z B A C Z B A C

Step 1 Step 2 Step 3 Step 4 Step 5

Slide 168 (Answer) / 183

82 What would be the description used to get from Step 4 to Step 5? A Construct an arc located in the interior of angle Z using a compass centered at point B with a radius length that is congruent to the radius length used to draw the arc centered at point A. Label the intersection point of the 2 interior arcs point C. B Construct an arc located in the interior of angle Z using a compass centered at point A and a radius greater than half of angle ZBA. C Draw a ray ZC, which is the angle bisector of angle BZA. D Construct an arc using a compass centered at point Z and any radius

• length. Label the points where the arc intersects the angle A and B.

Question 2/11

Topic: Angle Constructions (Unit 2) & Triangle Congruence Proofs Marcella drew each step of a construction of an angle bisector. Z B A Z Z B A Z B A C Z B A C

Step 1 Step 2 Step 3 Step 4 Step 5

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Answer

C

Slide 169 / 183

Part B Marcella wants to explain why the construction produces and angle

• bisector. She makes a new step with line segments AB and BC added

to the construction, as shown. Using the figure, prove that ray ZC bisects angle AZB. Be sure to justify each statement of your proof. This is a great problem and draws on a lot of what we've learned. Try it in your groups. Then we'll work on it step by step together by asking questions that break the problem into pieces.

Question 2/11

Topic: Angle Constructions (Unit 2) & Triangle Congruence Proofs

B Z C A

Slide 170 / 183

83 What have we learned that will help solve this problem?

A Construction of an angle bisector w/ a compass and straightedge B Ways to prove triangles congruent C The corresponding parts of congruent triangles are congruent (CPCTC) D All of the above B Z C A

Slide 170 (Answer) / 183

83 What have we learned that will help solve this problem?

A Construction of an angle bisector w/ a compass and straightedge B Ways to prove triangles congruent C The corresponding parts of congruent triangles are congruent (CPCTC) D All of the above B Z C A

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Answer

D All of the above

Slide 171 / 183

84 What should be the first statement in our proof?

A ZA ≅ ZB B ∠BZC ≅ ∠AZC C ∆BZC ≅ ∆AZC D ZC bisects ∠AZB B Z C A

Slide 171 (Answer) / 183

84 What should be the first statement in our proof?

A ZA ≅ ZB B ∠BZC ≅ ∠AZC C ∆BZC ≅ ∆AZC D ZC bisects ∠AZB B Z C A

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Answer

A

Slide 172 / 183

85 Why can we say that these two segments in step #1 are

congruent? A CPCTC B Definition of an Angle Bisector C Reflexive Property of Congruence D Both segments were drawn with the same compass setting, and all radii of a given circle are congruent. B Z C A

Slide 172 (Answer) / 183

85 Why can we say that these two segments in step #1 are

congruent? A CPCTC B Definition of an Angle Bisector C Reflexive Property of Congruence D Both segments were drawn with the same compass setting, and all radii of a given circle are congruent. B Z C A

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Answer

D

Slide 173 / 183

86 What should be the second statement in our proof?

A BC ≅ AC B ∠BZC ≅ ∠AZC C ∆BZC ≅ ∆AZC D ZC bisects ∠AZB B Z C A

Slide 173 (Answer) / 183

86 What should be the second statement in our proof?

A BC ≅ AC B ∠BZC ≅ ∠AZC C ∆BZC ≅ ∆AZC D ZC bisects ∠AZB B Z C A

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Answer

A

Slide 174 / 183

87 Why can we say that these two segments in step #2 are

congruent? A CPCTC B Definition of an Angle Bisector C Reflexive Property of Congruence D Both segments were drawn with the same compass setting, and all radii of a given circle are congruent. B Z C A

Slide 174 (Answer) / 183

87 Why can we say that these two segments in step #2 are

congruent? A CPCTC B Definition of an Angle Bisector C Reflexive Property of Congruence D Both segments were drawn with the same compass setting, and all radii of a given circle are congruent. B Z C A

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Answer

D

Slide 175 / 183

88 What should be the third statement in our proof?

A ZC ≅ ZC B ∠BZC ≅ ∠AZC C ∆BZC ≅ ∆AZC D ZC bisects ∠AZB B Z C A

Slide 175 (Answer) / 183

88 What should be the third statement in our proof?

A ZC ≅ ZC B ∠BZC ≅ ∠AZC C ∆BZC ≅ ∆AZC D ZC bisects ∠AZB B Z C A

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Answer

A

Slide 176 / 183

89 Why can we say that these two segments in step #3 are

congruent? A CPCTC B Definition of an Angle Bisector C Reflexive Property of Congruence D Both segments were drawn with the same compass setting, and all radii of a given circle are congruent. B Z C A

Slide 176 (Answer) / 183

89 Why can we say that these two segments in step #3 are

congruent? A CPCTC B Definition of an Angle Bisector C Reflexive Property of Congruence D Both segments were drawn with the same compass setting, and all radii of a given circle are congruent. B Z C A

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Answer

C

Slide 177 / 183

90 What should be the fourth statement in our proof?

A ZC ≅ ZC B ∠BZC ≅ ∠AZC C ∆BZC ≅ ∆AZC D ZC bisects ∠AZB B Z C A

Slide 177 (Answer) / 183

90 What should be the fourth statement in our proof?

A ZC ≅ ZC B ∠BZC ≅ ∠AZC C ∆BZC ≅ ∆AZC D ZC bisects ∠AZB B Z C A

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Answer

C

Slide 178 / 183

91 Why can we say that these two triangles in step #4 are

congruent? A SSS Triangle Congruence B SAS Triangle Congruence C AAS Triangle Congruence D ASA Triangle Congruence E HL Triangle Congruence B Z C A

Slide 178 (Answer) / 183

91 Why can we say that these two triangles in step #4 are

congruent? A SSS Triangle Congruence B SAS Triangle Congruence C AAS Triangle Congruence D ASA Triangle Congruence E HL Triangle Congruence B Z C A

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Answer

A

Slide 179 / 183

92 Since we know that the triangles are congruent, what

should be the fifth statement in our proof? A ZC ≅ ZC B ∠BZC ≅ ∠AZC C ∆BZC ≅ ∆AZC D ZC bisects ∠AZB B Z C A

Slide 179 (Answer) / 183

92 Since we know that the triangles are congruent, what

should be the fifth statement in our proof? A ZC ≅ ZC B ∠BZC ≅ ∠AZC C ∆BZC ≅ ∆AZC D ZC bisects ∠AZB B Z C A

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Answer

B

Slide 180 / 183

93 Why can we say that these two angles in step #5 are

congruent? A CPCTC B Definition of an Angle Bisector C Reflexive Property of Congruence D Both segments were drawn with the same compass setting, and all radii of a given circle are congruent. B Z C A

Slide 180 (Answer) / 183

93 Why can we say that these two angles in step #5 are

congruent? A CPCTC B Definition of an Angle Bisector C Reflexive Property of Congruence D Both segments were drawn with the same compass setting, and all radii of a given circle are congruent. B Z C A

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Answer

A

Slide 181 / 183

94 Since we know that the angles are congruent, what

should be the sixth statement in our proof? A ZC ≅ ZC B ∠BZC ≅ ∠AZC C ∆BZC ≅ ∆AZC D ZC bisects ∠AZB B Z C A

Slide 181 (Answer) / 183

94 Since we know that the angles are congruent, what

should be the sixth statement in our proof? A ZC ≅ ZC B ∠BZC ≅ ∠AZC C ∆BZC ≅ ∆AZC D ZC bisects ∠AZB B Z C A

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Answer

D

Slide 182 / 183

95 What is the final reason in our proof?

A CPCTC B Definition of an Angle Bisector C Reflexive Property of Congruence D Both segments were drawn with the same compass setting, and all radii of a given circle are congruent. B Z C A

Slide 182 (Answer) / 183

95 What is the final reason in our proof?

A CPCTC B Definition of an Angle Bisector C Reflexive Property of Congruence D Both segments were drawn with the same compass setting, and all radii of a given circle are congruent. B Z C A

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Answer

B

Slide 183 / 183

Statements Reasons 1) ZA ≅ ZB 1) Both segments were drawn with the same compass setting, and all radii of a given circle are congruent. 2) AC ≅ BC 2) Both segments were drawn with the same compass setting, and all radii of a given circle are congruent. 3) ZC ≅ ZC 3) Reflexive Property of ≅ 4) ∆AZC ≅ ∆BZC 4) SSS Triangle ≅ 5) ∠AZC ≅ ∠BZC 5) CPCTC 6) ZC is bisects ∠AZB 6) Definition of an Angle Bisector Given: The construction of the figure to the right Prove: ZC bisects ∠AZB Below is a completed version of the proof that we just wrote.

B Z C A