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Geometry

Triangles

2015-12-08 www.njctl.org

Slide 3 / 232

Table of Contents

· PARCC Sample Question and Applications · Triangle Sum Theorem · Exterior Angle Theorem · Triangles · Inequalities in Triangles · Similar Triangles

Click on the topic to go to that section

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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.

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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.

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Math Practice

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Return to Table

• f Contents

Triangles

Slide 6 / 232 Geometric Figures

Euclid now makes the transitions to geometric figures, which are created by a boundary which separates space into that which is within the figure and that which is not. Definition 13. A boundary is that which is an extremity of anything. Definition 14. A figure is that which is contained by any boundary

• r boundaries.

Slide 7 / 232 Geometric Figures

His definitions from 15 to 18 relate to circles, which we will discuss

• later. In this chapter, we will be discussing triangles, which are an

example of a rectilinear figure: a figure bounded by straight lines. A triangle is bounded by three lines. Definition 19. Rectilinear figures are those which are contained by straight lines, trilateral figures being those contained by three, quadrilateral those contained by four, and multilateral those contained by more than four straight lines.

Slide 8 / 232 Parts of a Triangle

Each triangle has three sides and three vertices. Each vertex is where two sides meet. A pair of sides and the vertex define an angle, so each triangle includes three angles. Write "side" next to each side and circle the vertices on the triangle below.

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1 The letter on this triangle that corresponds to a side is: A B C

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1 The letter on this triangle that corresponds to a side is: A B C

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Answer

C Slide 10 / 232

2 The letter on this triangle that represents a vertex is: A B C

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2 The letter on this triangle that represents a vertex is: A B C

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Answer

A Slide 11 / 232 Parts of a Triangle

C A B

Each vertex is named with a letter. The sides can then be named with the letters of the two vertices on either side of it. The triangle is named with a triangle symbol Δ in front followed by the three letters of its vertices. Name the 3 sides of this triangle ______ ______ ______

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3 What is the name of the side shown in red? A AB B BC C AC C A B

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3 What is the name of the side shown in red? A AB B BC C AC C A B

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Answer

C Slide 13 / 232

4 What is the name of the side shown in red? A AB B BC C AC C A B

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4 What is the name of the side shown in red? A AB B BC C AC C A B

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Answer

A

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5 Which of the following are names of this triangle? A ΔABC B ΔBCA C ΔACB C A B D ΔCAB E all of these

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5 Which of the following are names of this triangle? A ΔABC B ΔBCA C ΔACB C A B D ΔCAB E all of these

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Answer

E Slide 15 / 232 Parts of a Triangle

C A B

In the above, the red side is ________________ A, while the green sides are ________________ to A. A side is opposite an angle if it does not touch it. Otherwise, it is adjacent to the angle.

Slide 15 (Answer) / 232 Parts of a Triangle

C A B

In the above, the red side is ________________ A, while the green sides are ________________ to A. A side is opposite an angle if it does not touch it. Otherwise, it is adjacent to the angle.

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Answer

• pposite

adjacent Questions on this slide address MP6

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6 Which side is opposite angle B? A AB B CA C BC D None C A B

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6 Which side is opposite angle B? A AB B CA C BC D None C A B

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Answer

B

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7 Which side is opposite angle A? A AB B CA C BC D None C A B

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7 Which side is opposite angle A? A AB B CA C BC D None C A B

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Answer

C Slide 18 / 232

8 Which sides are adjacent to angle C? A AB & BC B CA & BA C BC & CA D None C A B

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8 Which sides are adjacent to angle C? A AB & BC B CA & BA C BC & CA D None C A B

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Answer

C Slide 19 / 232

9 Which sides are adjacent to angle B? A AB & BC B CA & BA C BC & CA D None C A B

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9 Which sides are adjacent to angle B? A AB & BC B CA & BA C BC & CA D None C A B

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Answer

A

Slide 20 / 232 Types of Triangles

In general, a triangle can have sides of all different lengths and angles

• f all different measure.

However, there are names given to triangles which have specific or special angles or some number of equal sides or angles. Euclid defined the names for a number of these in his definitions.

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Definition 20: Of trilateral figures, an equilateral triangle is that which has its three sides equal, an isosceles triangle is that which has two

• f its sides alone equal, and a scalene triangle is that which has its

three sides unequal

Classifying Triangles

Triangles can be classified by their sides or by their angles. In this definition, Euclid used the sides. In his next definition, Euclid uses the angles.

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Definition 21: Further, of trilateral figures, a right-angled triangle is that which has a right angle, an obtuse-angled triangle is that which has an obtuse angle, and an acute-angled triangle is that which has its three angles acute.

Classifying Triangles

We will draw from both definitions, since in several cases both definitions apply to the same triangle.

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Acute Triangles In an acute triangle, every angle of a triangle is acute. Notice that no angle is equal to or greater than 90º in this triangle.

Classifying Triangles

Definition 21: "...an acute-angled triangle is that which has its three angles acute."

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Right Triangles A right triangle has one right angle and two acute angles. Notice that one angle is 90º, which means that the other two sum to 90º; and they are acute. The side opposite the right angle is called the hypotenuse and the

• ther two sides are called the

legs.

Classifying Triangles

Definition 21: "...a right-angled triangle is that which has a right angle..."

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Isosceles Triangles An isosceles triangle has at least two sides with equal length. The angles opposite those equal sides are of equal measure.

xº xº

Classifying Triangles

Definition 20: "...an isosceles triangle is that which has two of its sides alone equal..."

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Isosceles Triangles The equal angles, of measure xº in this diagram, are called the base angles. The side between them is called the base. The other two sides, opposite the base angles and congruent to each other are called the legs. This is a special case of an acute triangle.

xº xº

Classifying Triangles Slide 27 / 232

Obtuse Triangles An obtuse triangle has one angle which is greater than 90 º and two acute angles. Notice that one angle is greater than 90º, which means that the other two sum to less than 90º; and they are acute..

Classifying Triangles

Definition 21: "...an obtuse-angled triangle is that which has an obtuse angle..."

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Equiangular / Equilateral Triangles An equiangular, or equilateral, triangle has angles of equal measure and sides of equal length. Definition 20: "...an equilateral triangle is that which has its three sides equal..." All the angles are of equal measure and all the sides are of equal length. Each angle measures 60º. This is a special acute isosceles triangle.

xº xº xº

Classifying Triangles Slide 29 / 232 Classifying Triangles

Scalene Triangles None of the sides or angles of a scalene triangle are congruent with one another. Definition 20: "...a scalene triangle is that which has its three sides unequal..." Note that in this triangle none of the sides or angles are equal.

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10 An isosceles triangle is _______________ an equilateral triangle. A Sometimes B Always C Never

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10 An isosceles triangle is _______________ an equilateral triangle. A Sometimes B Always C Never

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Answer

A

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11 An obtuse triangle is _______________ an isosceles triangle. A Sometimes B Always C Never

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11 An obtuse triangle is _______________ an isosceles triangle. A Sometimes B Always C Never

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Answer

A Slide 32 / 232

12 A triangle can have more than one obtuse angle. True False

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12 A triangle can have more than one obtuse angle. True False

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Answer

False Slide 33 / 232

13 A triangle can have more than one right angle. True False

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13 A triangle can have more than one right angle. True False

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Answer

False

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14 Each angle in an equiangular triangle measures 60° True False

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14 Each angle in an equiangular triangle measures 60° True False

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Answer

True Slide 35 / 232

15 An equilateral triangle is also an isosceles triangle True False

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15 An equilateral triangle is also an isosceles triangle True False

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Answer

False Slide 36 / 232

16 This triangle is classified as _____. (Choose all that apply.) A acute B right C isosceles D obtuse E equilateral F equiangular G scalene 60º 8.6 60º 60º 8.6 8.6

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16 This triangle is classified as _____. (Choose all that apply.) A acute B right C isosceles D obtuse E equilateral F equiangular G scalene 60º 8.6 60º 60º 8.6 8.6

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Answer

A, C, E, F

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17 This triangle is classified as _____. (Choose all that apply.) A acute B right C isosceles D obtuse E equilateral F equiangular G scalene 57º 79º 44º 6.1 8.7 7.4

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17 This triangle is classified as _____. (Choose all that apply.) A acute B right C isosceles D obtuse E equilateral F equiangular G scalene 57º 79º 44º 6.1 8.7 7.4

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Answer

A, G Slide 38 / 232

18 This triangle is classified as _____. (Choose all that apply.) A acute B right C isosceles D obtuse E equilateral F equiangular G scalene 26° 128° 26° 2.5 2.5 4.5

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18 This triangle is classified as _____. (Choose all that apply.) A acute B right C isosceles D obtuse E equilateral F equiangular G scalene 26° 128° 26° 2.5 2.5 4.5

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Answer

C, D Slide 39 / 232

19 This triangle is classified as _____. Choose all that apply. A acute B right C isosceles D obtuse E equilateral F equiangular G scalene 4.8 4.8 45° 45° 6.8

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19 This triangle is classified as _____. Choose all that apply. A acute B right C isosceles D obtuse E equilateral F equiangular G scalene 4.8 4.8 45° 45° 6.8

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Answer

B, C

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Measure and Classify the triangle by sides and angles

Example

isosceles, acute

Click for Answer Click for Answer

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Measure and Classify the triangle by sides and angles

Example

isosceles, acute

Click for Answer Click for Answer

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Math Practice

This example addresses MP6 Additional Q's to address MP standards: What is the question asking? (MP1) How could you prove that your answer is correct? (MP3) What tools could you use to solve this problem? (MP5)

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Measure and Classify the triangle by sides and angles

Example

scalene, obtuse

Click for Answer Click for Answer

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Measure and Classify the triangle by sides and angles

Example

scalene, obtuse

Click for Answer Click for Answer

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Math Practice

This example addresses MP6 Additional Q's to address MP standards: What is the question asking? (MP1) How could you prove that your answer is correct? (MP3) What tools could you use to solve this problem? (MP5)

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Measure and Classify the triangle by sides and angles

Example

scalene, acute

Click for Answer Click for Answer

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Measure and Classify the triangle by sides and angles

Example

scalene, acute

Click for Answer Click for Answer

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Math Practice

This example addresses MP6 Additional Q's to address MP standards: What is the question asking? (MP1) How could you prove that your answer is correct? (MP3) What tools could you use to solve this problem? (MP5)

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20 Classify the triangle with the given information: Side lengths: 3 cm, 4 cm, 5 cm A Equilateral B Isosceles C Scalene D Acute E Equiangular F Right G Obtuse

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20 Classify the triangle with the given information: Side lengths: 3 cm, 4 cm, 5 cm A Equilateral B Isosceles C Scalene D Acute E Equiangular F Right G Obtuse

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Answer

C Scalene Bonus: F Right 32+42 = 52 9+16 = 25

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21 Classify the triangle with the given information: Side lengths: 3 cm, 2 cm, 3 cm A Equilateral B Isosceles C Scalene D Acute E Equiangular F Right G Obtuse

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21 Classify the triangle with the given information: Side lengths: 3 cm, 2 cm, 3 cm A Equilateral B Isosceles C Scalene D Acute E Equiangular F Right G Obtuse

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Answer

B Isosceles Bonus: D Acute

3 cm 3 cm 2 cm

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22 Classify the triangle with the given information: Side lengths: 5 cm, 5 cm, 5 cm A Equilateral B Isosceles C Scalene D Acute E Equiangular F Right G Obtuse

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22 Classify the triangle with the given information: Side lengths: 5 cm, 5 cm, 5 cm A Equilateral B Isosceles C Scalene D Acute E Equiangular F Right G Obtuse

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Answer

A Equilateral B Isosceles Bonus: E Equiangular (all equilateral triangles are equiangular) D Acute (all angles are 60°)

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23 Classify the triangle with the given information: Angle Measures: 25°, 120°, 35° A Equilateral B Isosceles C Scalene D Acute E Equiangular F Right G Obtuse

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23 Classify the triangle with the given information: Angle Measures: 25°, 120°, 35° A Equilateral B Isosceles C Scalene D Acute E Equiangular F Right G Obtuse

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Answer

G Obtuse Bonus: C Scalene (all angles are different, so all sides are different)

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24 Classify the triangle with the given information: Angle Measures: 30°, 60°, 90° A Equilateral B Isosceles C Scalene D Acute E Equiangular F Right G Obtuse

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24 Classify the triangle with the given information: Angle Measures: 30°, 60°, 90° A Equilateral B Isosceles C Scalene D Acute E Equiangular F Right G Obtuse

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Answer

F Right Bonus: C Scalene (all angles are different, so all sides are different)

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25 Classify the triangle with the given information: Side lengths: 3 cm, 4 cm, 5 cm Angle measures: 37°, 53°, 90° A Equilateral B Isosceles C Scalene D Acute E Equiangular F Right G Obtuse

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25 Classify the triangle with the given information: Side lengths: 3 cm, 4 cm, 5 cm Angle measures: 37°, 53°, 90° A Equilateral B Isosceles C Scalene D Acute E Equiangular F Right G Obtuse

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Answer

C Scalene F Right

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26 Classify the triangle by sides and angles A Equilateral B Isosceles C Scalene D Acute E Equiangular F Right G Obtuse A B 120° C

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26 Classify the triangle by sides and angles A Equilateral B Isosceles C Scalene D Acute E Equiangular F Right G Obtuse A B 120° C

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Answer

C Scalene G Obtuse

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L M N 27 Classify the triangle by sides and angles A Equilateral B Isosceles C Scalene D Acute E Equiangular F Right G Obtuse

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L M N 27 Classify the triangle by sides and angles A Equilateral B Isosceles C Scalene D Acute E Equiangular F Right G Obtuse

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Answer

B Isosceles F Right

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H J K 45° 85° 50° 28 Classify the triangle by sides and angles A Equilateral B Isosceles C Scalene D Acute E Equiangular F Right G Obtuse

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H J K 45° 85° 50° 28 Classify the triangle by sides and angles A Equilateral B Isosceles C Scalene D Acute E Equiangular F Right G Obtuse

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Answer

C Scalene D Acute

Slide 52 / 232

Triangle Sum Theorem

Return to Table

• f Contents

Slide 53 / 232 Triangle Sum Theorem

A B C

We can use what we learned about parallel lines to determine the sum of the measures of the angles of any triangle. First, let's draw two parallel lines. The first along the base of the triangle and the other through the opposite vertex.

Slide 53 (Answer) / 232 Triangle Sum Theorem

A B C

We can use what we learned about parallel lines to determine the sum of the measures of the angles of any triangle. First, let's draw two parallel lines. The first along the base of the triangle and the other through the opposite vertex.

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Math Practice

This example on this slide and the next 5 slides address MP2, MP3, MP4, MP5, and MP7

Slide 54 / 232

And extend AB to make it a transversal. Then, let's label some of the angles.

Triangle Sum Theorem

A B C x x y y

Slide 55 / 232

29 What is the name for the pair of angles labeled x and what is the relationship between them? A

• utside exterior, they are unequal

B alternate interior, they are unequal C alternate interior, they are equal D

• utside exterior, they are equal

Is the same true for the pair of angles labeled y?

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29 What is the name for the pair of angles labeled x and what is the relationship between them? A

• utside exterior, they are unequal

B alternate interior, they are unequal C alternate interior, they are equal D

• utside exterior, they are equal

Is the same true for the pair of angles labeled y?

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Answer

C; yes

Slide 56 / 232

A B C Therefore, both angles labeled x are equal and can be called x, and x has the same measure as B. x x Repeat the same process with side AC and find an angle along the upper parallel line equal to angle C

Triangle Sum Theorem Slide 57 / 232

A B C x x y y Let's just re-label the upper angles with A, B and C.

Triangle Sum Theorem Slide 58 / 232

A B C The sum of those angles along that upper parallel line equals 180º, so A + B + C = 180º B C We made no special assumptions about this triangle, so this proof applies to all triangles: the sum of the interior angles of any triangle is 180º

Triangle Sum Theorem Slide 59 / 232

The measures of the interior angles of a triangle sum to 180°

Click here to go to the lab titled, "Triangle Sum Theorem"

Triangle Sum Theorem

A B C

Slide 59 (Answer) / 232

The measures of the interior angles of a triangle sum to 180°

Click here to go to the lab titled, "Triangle Sum Theorem"

Triangle Sum Theorem

A B C

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Math Practice

This lab addresses MP4

Slide 60 / 232 Example: Triangle Sum Theorem

32º J K L 20º Find the measure of the missing angle.

Slide 60 (Answer) / 232 Example: Triangle Sum Theorem

32º J K L 20º Find the measure of the missing angle.

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Answer m∠J + m∠K + m∠L = 180º 32º + m∠K + 20 = 180º m∠K + 52º = 180º m∠K = 128º

This example addresses MP2. Additional Q's that address MP's: How are the angles in a triangle related to each other? (MP7) What information are you given? (MP1) Create an equation to represent the problem. (MP2)

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30 What is m∠B? A B C 52° 53°

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30 What is m∠B? A B C 52° 53°

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Answer

52 + 53 + m ∠B = 180° 105 + m∠B = 180°

• 105 -105

m∠B = 75°

Slide 62 / 232

31 What is the measurement of the missing angle? 57° L M N

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31 What is the measurement of the missing angle? 57° L M N

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Answer

90 + 57 + m ∠N = 180° 147 + m∠N = 180°

• 147 -147

m∠N = 33°

Slide 63 / 232

32 In ΔABC, if m∠B is 84° and m∠C is 36°, what is m∠A?

Slide 63 (Answer) / 232

32 In ΔABC, if m∠B is 84° and m∠C is 36°, what is m∠A?

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Answer

triangle ABC = 180° So, 84 + 36 + m∠A = 180° m∠A =180° - 120° m∠A = 60°

Slide 64 / 232

33 In ΔDEF, if m∠D is 63° and m∠E is 12°, find m∠F.

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33 In ΔDEF, if m∠D is 63° and m∠E is 12°, find m∠F.

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Answer

triangle DEF = 180° So, 63 + 12 + m ∠F = 180° m∠F =180° - 75° m∠F = 105°

Slide 65 / 232

Solve for x

55° (12x+8)° (8x-3)° P Q R

Example Slide 65 (Answer) / 232

Solve for x

55° (12x+8)° (8x-3)° P Q R

Example

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Answer m∠P + m∠Q + m∠R = 180º 55º + (12x+8) + (8x-3) = 180º 20x +60 = 180 20x = 120 x = 6

This example addresses MP2. Additional Q's that address MP's: How are the angles in a triangle related to each other? (MP7) What information are you given? (MP1) Create an equation to represent the problem. (MP2)

Slide 66 / 232

Q R S 2x° 5x° 8x° 34 Solve for x. Then find: m∠Q = m∠ R = m∠S =

Slide 66 (Answer) / 232

Q R S 2x° 5x° 8x° 34 Solve for x. Then find: m∠Q = m∠ R = m∠S =

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Answer

2x+5x+8x = 180 15x = 180 x = 12 m∠Q = 24° m∠R = 96° m∠S = 60° Extension Answer

Slide 67 / 232

35 What is the measure of ∠B? C B A (3x-17)° (x+40)° (2x-5)°

Slide 67 (Answer) / 232

35 What is the measure of ∠B? C B A (3x-17)° (x+40)° (2x-5)°

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Answer

3x-17+2x-5+x+40 = 180 6x + 18 = 180 6x = 162 x = 27 m∠B = 3(27)-17 = 81 - 17 = 64

Slide 68 / 232 Corollary to Triangle Sum Theorem

The acute angles of a right triangle are complementary. A B C

Slide 69 / 232

Given: Triangle ABC is a right triangle Prove: Its acute angles, Angles B and C, are complementary A B C

Proof of Triangle Sum Theorem Corollary Slide 70 / 232

36 Which reason applies to step 1? A Subtraction Property of Equality B Substitution Property of Equality C Given D Definition of right triangle E Definition of a right angle A B C

Statement Reason 1 Triangle ABC is a right triangle ? 2 Right triangles contain a right angle. ? 3 ? Interior Angles Theorem 4 m∠A = 90º ? 5 90º + m∠B + m∠C = 180º ? 6 m∠B + m∠C = 90º ? 7 ? Definition of complementary Answer

Slide 71 / 232

37 Which reason applies to step 2? A Subtraction Property of Equality B Substitution Property of Equality C Given D Definition of right triangle E Definition of a right angle A B C

Statement Reason 1 Triangle ABC is a right triangle ? 2 Right triangles contain a right angle. ? 3 ? Interior Angles Theorem 4 m∠A = 90º ? 5 90º + m∠B + m∠C = 180º ? 6 m∠B + m∠C = 90º ? 7 ? Definition of complementary Answer

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38 Which reason applies to step 3? A B C A The measure of a straight angle is 180º B m∠A + m∠B + m∠C = 180º C m∠B + m∠C = 90º D m∠B + m∠C = 180º E ∠A is a right angle

Statement Reason 1 Triangle ABC is a right triangle ? 2 Right triangles contain a right angle. ? 3 ? Interior Angles Theorem 4 m∠A = 90º ? 5 90º + m∠B + m∠C = 180º ? 6 m∠B + m∠C = 90º ? 7 ? Definition of complementary Answer

Slide 73 / 232

39 Which reason applies to step 4? A Subtraction Property of Equality B Substitution Property of Equality C Given D Definition of right triangle E Definition of a right angle A B C

Statement Reason 1 Triangle ABC is a right triangle ? 2 Right triangles contain a right angle. ? 3 ? Interior Angles Theorem 4 m∠A = 90º ? 5 90º + m∠B + m∠C = 180º ? 6 m∠B + m∠C = 90º ? 7 ? Definition of complementary Answer

Slide 74 / 232

40 Which reason applies to step 5? A Subtraction Property of Equality B Substitution Property of Equality C Given D Definition of right triangle E Definition of a right angle A B C

Statement Reason 1 Triangle ABC is a right triangle ? 2 Right triangles contain a right angle. ? 3 ? Interior Angles Theorem 4 m∠A = 90º ? 5 90º + m∠B + m∠C = 180º ? 6 m∠B + m∠C = 90º ? 7 ? Definition of complementary Answer

Slide 75 / 232

41 Which reason applies to step 6? A Subtraction Property of Equality B Substitution Property of Equality C Given D Definition of right triangle E Definition of a right angle A B C

Statement Reason 1 Triangle ABC is a right triangle ? 2 Right triangles contain a right angle. ? 3 ? Interior Angles Theorem 4 m∠A = 90º ? 5 90º + m∠B + m∠C = 180º ? 6 m∠B + m∠C = 90º ? 7 ? Definition of complementary Answer

Slide 76 / 232

42 Which reason applies to step 7? A B C

Statement Reason 1 Triangle ABC is a right triangle ? 2 Right triangles contain a right angle. ? 3 ? Interior Angles Theorem 4 m∠A = 90º ? 5 90º + m∠B + m∠C = 180º ? 6 m∠B + m∠C = 90º ? 7 ? Definition of complementary

A The measure of a straight angle is 180º B The sum of the interior angles of a triangle is 180º C The acute angles are complementary D The acute angles are supplementary E ∠A is a right angle

Answer

Slide 77 / 232

A B C Given: Triangle ABC is a right triangle Prove: Its acute angles, Angles B and C, are complementary Statement Reason 1 Triangle ABC is a right triangle Given 2 Right triangles contain a right angle. Definition of right triangle 3 m∠A + m∠B + m∠C = 180º Interior Angles Theorem 4 m∠A = 90º Definition of right angle 5 90º + m∠B + m∠C = 180º Substitution Property of Equality 6 m∠B + m∠C = 90º Subtraction Property of Equality 7 The acute angles are complementary Definition of complementary

Proof of Triangle Sum Theorem Corollary Slide 78 / 232 Example

The measure of one acute angle of a right triangle is five times the measure of the other acute angle. Find the measure of each acute angle.

Slide 78 (Answer) / 232 Example

The measure of one acute angle of a right triangle is five times the measure of the other acute angle. Find the measure of each acute angle.

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Answer x + 5x + 90 = 180º 6x = 90 x = 15 The measures of the angles are 15º and 75º.

This example addresses MP2. Additional Q's that address MP's: How are the angles in a triangle related to each other? (MP7) What information are you given? (MP1) Create an equation to represent the problem. (MP2)

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43 In a right triangle, the two acute angles sum to 90°. True False

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43 In a right triangle, the two acute angles sum to 90°. True False

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Answer

True Slide 80 / 232

44 What is the measurement of the missing angle? 57° L M N

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44 What is the measurement of the missing angle? 57° L M N

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Answer

x+57 = 90

º

x = 33

º

Note: we solved this problem earlier using the Triangle Sum Theorem. Use the Corollary to the Triangle Sum this time.

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45 Solve for x. A B C C B A What are the measures of the three angles?

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45 Solve for x. A B C C B A What are the measures of the three angles?

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Answer

3x-1+31 = 90 3x + 30 = 90 3x = 60 x = 20 Challenge Answer m∠A = 59

º

m∠B = 90º m∠C = 31º

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46 Solve for x. What are the measures of the three angles?

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46 Solve for x. What are the measures of the three angles?

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Answer

2x-2+x+5 = 90 3x + 3 = 90 3x = 87 x = 29 Challenge Answer m∠D = 90º m∠E = 56º m∠F = 34º

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47 m∠1 + m∠2 = 1 2 3

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47 m∠1 + m∠2 = 1 2 3

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Answer

90 Slide 84 / 232

48 m∠1 + m∠3 = 1 2 3

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48 m∠1 + m∠3 = 1 2 3

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Answer

90 Slide 85 / 232

20° x° 49 Find the value of x in the diagram

Slide 85 (Answer) / 232

20° x° 49 Find the value of x in the diagram

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Exterior Angle Theorem

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• f Contents

Slide 87 / 232

Exterior angles are formed by extending any side of a triangle. The exterior angle is then the angle between that extended side and the nearest side of the triangle. One exterior angle is shown below. Take a moment and draw another.

Exterior Angles

A B C xº

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Since a triangle has three vertices and two external angles can be drawn at each vertex, it is possible to draw six external angles to a triangle. Draw the other external angle at Vertex A.

Exterior Angles

A B C xº

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A B C xº xº The exterior angles at each vertex are congruent, since they are vertical angles.

Exterior Angles Slide 90 / 232

The interior angles of this triangle are ∠A, ∠ABC and ∠C. Once an exterior angle is drawn, one interior angle is adjacent, and the two others are remote. Since you can draw exterior angles at any vertex, any interior angle can be the remote depending on at which vertex you draw the external angle.

Remote Interior Angles

A B C In this case, ∠A and ∠C are the remote interior angles and ∠ABC is the adjacent interior angle. xº

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50 Which are the remote interior angles in this instance? A ∠A & ∠B B ∠A & ∠C C ∠B & ∠C A B C

xº xº

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50 Which are the remote interior angles in this instance? A ∠A & ∠B B ∠A & ∠C C ∠B & ∠C A B C

xº xº

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Answer

C

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51 If line AB is a straight line, what is the sum of ∠2 and ∠1? 1 A B 2

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51 If line AB is a straight line, what is the sum of ∠2 and ∠1? 1 A B 2

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Answer Linear Pair Postulate Since ∠1 & ∠2 are adjacent and lie

• n the same line, they are a linear

pair, which makes them

• supplementary. Since

supplementary angles sum to 180º, these two angles must also sum to 180º. In this diagram, m∠1 + m∠2 = 180º

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52 In this diagram, what is the sum of angles P, Q and R? P R Q

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52 In this diagram, what is the sum of angles P, Q and R? P R Q

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Answer

The sum of the angles of a triangle is 180º

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A B C D The measure of any exterior angle of a triangle is equal to the sum of its remote interior angles. m∠DBA = m∠A + m∠C

• r

x = m∠A + m∠C

Exterior Angles Theorem

xº

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Given: ∠DBA is an exterior angle of ΔABC and ∠A and ∠C are remote interior angles. Prove: m∠DBA = m∠A + m∠C

Proof of Exterior Angles Theorem

A B C D xº

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53 Which reason applies to step 2? A Angles that form a linear pair are supplementary B Definition of complementary C Interior Angles Theorem D Substitution Property of Equality E Definition of a right angle

A B C D xº

Statement Reason 1 ∠DBA is an exterior angle of ΔABC and ∠A and ∠C are remote interior angles Given 2 ∠DBA and ∠ABC are supplementary ? 3 ? Definition of supplementary 4 m∠A+ m∠ABC + m∠C = 180° ? 5 m∠DBA + m∠ABC = m∠A + m∠ABC + m∠C ? 6 ? Subtraction Property of Equality

Answer

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54 Which statement applies to step 3? A m∠DBA + m∠ABC = 180° B m∠DBA = m∠A + m∠C C m∠A + m∠B = 180° D m∠DBA + m∠A = 90° E m∠DBA + m∠A = 180°

A B C x D

Statement Reason 1 ∠DBA is an exterior angle of ΔABC and ∠A and ∠C are remote interior angles Given 2 ∠DBA and ∠ABC are supplementary ? 3 ? Definition of supplementary 4 m∠A+ m∠ABC + m∠C = 180° ? 5 m∠DBA + m∠ABC = m∠A + m∠ABC + m∠C ? 6 ? Subtraction Property of Equality

Answer

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55 Which reason applies to step 4? A Angles that form a linear pair are supplementary B Definition of complementary C Interior Angles Theorem D Substitution Property of Equality E Definition of a right angle

A B C x D

Statement Reason 1 ∠DBA is an exterior angle of ΔABC and ∠A and ∠C are remote interior angles Given 2 ∠DBA and ∠ABC are supplementary ? 3 ? Definition of supplementary 4 m∠A+ m∠ABC + m∠C = 180° ? 5 m∠DBA + m∠ABC = m∠A + m∠ABC + m∠C ? 6 ? Subtraction Property of Equality

Answer

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56 Which reason applies to step 5? A Angles that form a linear pair are supplementary B Definition of complementary C Interior Angles Theorem D Substitution Property of Equality E Definition of a right angle

A B C x D

Statement Reason 1 ∠DBA is an exterior angle of ΔABC and ∠A and ∠C are remote interior angles Given 2 ∠DBA and ∠ABC are supplementary ? 3 ? Definition of supplementary 4 m∠A+ m∠ABC + m∠C = 180° ? 5 m∠DBA + m∠ABC = m∠A + m∠ABC + m∠C ? 6 ? Subtraction Property of Equality

Answer

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57 Which statement applies to step 6? A m∠DBA + m∠ABC = 180° B m∠DBA = m∠A + m∠C C m∠A + m∠B = 180° D m∠DBA + m∠A = 90° E m∠DBA + m∠A = 180°

A B C x D

Statement Reason 1 ∠DBA is an exterior angle of ΔABC and ∠A and ∠C are remote interior angles Given 2 ∠DBA and ∠ABC are supplementary ? 3 ? Definition of supplementary 4 m∠A+ m∠ABC + m∠C = 180° ? 5 m∠DBA + m∠ABC = m∠A + m∠ABC + m∠C ? 6 ? Subtraction Property of Equality

Answer

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Statement Reason 1 ∠DBA is an exterior angle of ΔABC and ∠A and ∠C are remote interior angles Given 2 ∠DBA and ∠ABC are supplementary Angles that form a linear pair are supplementary 3 ∠DBA + m∠ABC = 180° Definition of supplementary 4 m∠A+ m∠ABC + m∠C = 180° Interior Angles Theorem 5 m∠DBA + m∠ABC = m∠A + m∠ABC + m∠C Substitution Property of Equality 6 m∠DBA = m∠A + m∠C Subtraction Property of Equality

Proof of Exterior Angles Theorem

Given: ∠DBA is an exterior angle

• f ΔABC and ∠A and ∠C are

remote interior angles. Prove: m∠DBA = m∠A + m∠C

A B C x D

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58 In this case, what must be the relationship between the interior angles of ΔPQR and ∠1? A m∠Q = m∠1 B m∠1 = m∠P C m∠1 = m∠Q + m∠R D m∠1 = m∠P + m∠R E m∠1 = m∠Q + m∠P

1

P R Q

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58 In this case, what must be the relationship between the interior angles of ΔPQR and ∠1? A m∠Q = m∠1 B m∠1 = m∠P C m∠1 = m∠Q + m∠R D m∠1 = m∠P + m∠R E m∠1 = m∠Q + m∠P

1

P R Q

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Answer

D Slide 103 / 232

59 In this case, what must be the relationship between the interior angles of ΔPQR and ∠2? A m∠Q = m∠2 B m∠2 = m∠P C m∠2 = m∠Q + m∠R D m∠2 = m∠P + m∠R E m∠2 = m∠Q + m∠P

2

P R Q

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59 In this case, what must be the relationship between the interior angles of ΔPQR and ∠2? A m∠Q = m∠2 B m∠2 = m∠P C m∠2 = m∠Q + m∠R D m∠2 = m∠P + m∠R E m∠2 = m∠Q + m∠P

2

P R Q

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Answer

C Slide 104 / 232

Example: Using the Exterior Angle Theorem

140º xº xº P Q R

What is the value of x?

Slide 104 (Answer) / 232

Example: Using the Exterior Angle Theorem

140º xº xº P Q R

What is the value of x?

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Answer

140

• = x
• + x
• 140 = 2x

70 = x The measure of the exterior angle is equal to the sum of the two angles that are not adjacent to the exterior angle.

Slide 105 / 232 Example

Solve for x and y.

21° 34° x° y°

Slide 105 (Answer) / 232 Example

Solve for x and y.

21° 34° x° y°

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Answer Exterior Angles Theorem x = 21 + 34 x = 55º Triangle Sum Theorem 21 + 34 + y = 180º 55 + y = 180

• 55
• 55

y = 125º

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xº yº 75º 50º

Example

Solve for x and y.

Slide 106 (Answer) / 232

xº yº 75º 50º

Example

Solve for x and y.

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Answer

Exterior Angles Theorem x = 75 + 50 x = 125º Triangle Sum Theorem 75 + 50 + y = 180 125 + y = 180

• 125
• 125

y = 55º Additional Q's that address MP Standards: What information are you given? (MP1) What do you need to find? (MP1) How is the variable related to the other angles in the diagram? (MP7) Why can you write that equation? (MP3) Can you find a shortcut to solve for y? How would your shortcut make the problem easier? (MP8)

• Ans: After finding x, if you subtract it from 180,

you calculate the value of y.

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60 Solve for x. xº yº 60º 55º

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60 Solve for x. xº yº 60º 55º

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Answer

65º

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61 Solve for y. xº yº 60º 55º

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61 Solve for y. xº yº 60º 55º

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Answer

115 º Slide 109 / 232

62 Find the value of x. 2xº yº 60º 94º

Slide 109 (Answer) / 232

62 Find the value of x. 2xº yº 60º 94º

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Answer

60 + 2x = 94 º 2x = 34 x = 17

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63 Find the value of x. (2x+3)º yº 100º 51º

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63 Find the value of x. (2x+3)º yº 100º 51º

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Answer

51 + 2x + 3 = 100 º 2x + 54 = 100 2x = 46 x = 23

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64 Find the value of x. (x+2)° y° (3x-5)° 33°

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64 Find the value of x. (x+2)° y° (3x-5)° 33°

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Answer

33 + x + 2 = 3x - 5 x + 35 = 3x - 5 35 = 2x - 5 40 = 2x 20 = x

Slide 112 / 232

65 Segment PS bisects ∠RST, what is the value of w? 25° P S T R wº

Slide 112 (Answer) / 232

65 Segment PS bisects ∠RST, what is the value of w? 25° P S T R wº

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Answer 25o 115

• P

R T

2 5

• 65o

S

Slide 113 / 232 Example

Find the missing angles in the diagram.

60° 7 103° 43° 45° 30° 5 4 3 2 1

Slide 113 (Answer) / 232 Example

Find the missing angles in the diagram.

60° 7 103° 43° 45° 30° 5 4 3 2 1

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Teacher Notes

Find the measures of all angles together m∠1 = 45o m∠2 = 90o m∠3 = 60o m∠4 = 60o m∠5 = 77o m∠6 = 77o m∠7 = 43o

MP's addressed: MP2

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40º 1 2 4 5 3 60º 66 Find the measure of ∠1.

Slide 114 (Answer) / 232

40º 1 2 4 5 3 60º 66 Find the measure of ∠1.

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Answer

40 + m∠1 = 90º m∠1 = 50º

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67 Find the measure of ∠2. 40º 1 2 4 5 3 60º

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67 Find the measure of ∠2. 40º 1 2 4 5 3 60º

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Answer

40 + m∠2 = 180º m∠2 = 140º

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68 Find the measure of ∠3. 40º 1 2 4 5 3 60º

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68 Find the measure of ∠3. 40º 1 2 4 5 3 60º

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Answer

40 + m∠3 = 180º m∠3 = 140º

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69 40º 1 2 4 5 3 60º Find the measure of ∠4.

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69 40º 1 2 4 5 3 60º Find the measure of ∠4.

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Answer

m∠4 = 40º vertical angles are congruent

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70 Find the measure of ∠5. 40º 1 2 4 5 3 60º

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70 Find the measure of ∠5. 40º 1 2 4 5 3 60º

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Answer

m∠4 = 40º Interior angles of a triangle add up to 180 º, so 40+60+m∠5 = 180º 100+m∠5 = 180º m∠5 = 80o

Slide 119 / 232

Inequalities in Triangles

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• f Contents

Slide 120 / 232 Inequalities in one Triangle

To investigate inequalities in one triangle download the sketch, "inequalities in one triangle" and the worksheet, "inequalities in

• ne triangle"

Go to the sketch, "Inequalities in one triangle." Go to the worksheet, "Inequalities in one triangle."

Slide 120 (Answer) / 232 Inequalities in one Triangle

To investigate inequalities in one triangle download the sketch, "inequalities in one triangle" and the worksheet, "inequalities in

• ne triangle"

Go to the sketch, "Inequalities in one triangle." Go to the worksheet, "Inequalities in one triangle."

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Math Practice

This lab addresses MP4, MP5 & MP8

Slide 121 / 232 Angle Inequalities in a Triangle

The longest side is always opposite the largest angle. The shortest side is always opposite the smallest angle.

Slide 122 / 232

71 Name the longest side of this triangle. A AB B BC C CA D They are all equal A B C 35° 60° 85°

Slide 122 (Answer) / 232

71 Name the longest side of this triangle. A AB B BC C CA D They are all equal A B C 35° 60° 85°

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Answer

A Slide 123 / 232

72 Name the shortest side of this triangle. A AB B BC C CA D They are all equal A B C 35° 60° 85°

Slide 123 (Answer) / 232

72 Name the shortest side of this triangle. A AB B BC C CA D They are all equal A B C 35° 60° 85°

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Answer

B

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73 Name the shortest side of this triangle. A AB B BC C CA D They are all equal A B C 35° 105° 40°

Slide 124 (Answer) / 232

73 Name the shortest side of this triangle. A AB B BC C CA D They are all equal A B C 35° 105° 40°

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Answer

B Slide 125 / 232

74 Name the largest angle of this triangle. A ∠A B ∠B C ∠C D They are all equal A B C 10 14 8

Slide 125 (Answer) / 232

74 Name the largest angle of this triangle. A ∠A B ∠B C ∠C D They are all equal A B C 10 14 8

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Answer

B Slide 126 / 232

75 Name the smallest angle of this triangle. A ∠A B ∠B C ∠C D They are all equal A B C 10 14 8

Slide 126 (Answer) / 232

75 Name the smallest angle of this triangle. A ∠A B ∠B C ∠C D They are all equal A B C 10 14 8

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Answer

A

Slide 127 / 232

A C 10 10 10 76 Name the smallest angle of this triangle. A ∠A B ∠B C ∠C D They are all equal B

Slide 127 (Answer) / 232

A C 10 10 10 76 Name the smallest angle of this triangle. A ∠A B ∠B C ∠C D They are all equal B

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Answer

D Slide 128 / 232 Length Inequalities in a Triangle

No side can be longer than the sum of the other two sides. No side can be less than the difference of the other two sides.

Slide 129 / 232 Length Inequalities in a Triangle

No side can be longer than the sum of the other two sides. This follows from the fact that if the two shorter sides cannot be placed at a 180º angle and exceed the length of the longest side, a triangle cannot be made. As shown below, if the blue side is longer than the sum of the red and the green side, it cannot form a triangle. Move the sides below and try to form a triangle.

Slide 129 (Answer) / 232 Length Inequalities in a Triangle

No side can be longer than the sum of the other two sides. This follows from the fact that if the two shorter sides cannot be placed at a 180º angle and exceed the length of the longest side, a triangle cannot be made. As shown below, if the blue side is longer than the sum of the red and the green side, it cannot form a triangle. Move the sides below and try to form a triangle.

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Math Practice

The task at the bottom of this slide addresses MP4, MP5 & MP7

Slide 130 / 232 Length Inequalities in a Triangle

No side can be less than the difference of the other two sides. This follows from the fact that if the longer sides cannot, when placed at a 0° angle, reach the end of the shorter side, a triangle cannot be made. As shown below, if the blue side is too short to reach the red line, even when the red line is at the smallest angle, it cannot form a triangle.

Slide 130 (Answer) / 232 Length Inequalities in a Triangle

No side can be less than the difference of the other two sides. This follows from the fact that if the longer sides cannot, when placed at a 0° angle, reach the end of the shorter side, a triangle cannot be made. As shown below, if the blue side is too short to reach the red line, even when the red line is at the smallest angle, it cannot form a triangle.

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Math Practice

This slide addresses MP4, MP5 & MP7

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77 What is the maximum length of the third side to form a triangle if the other sides are 4 and 6?

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77 What is the maximum length of the third side to form a triangle if the other sides are 4 and 6?

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Answer

third side < 4 + 6 third side < 10

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78 What is the maximum length of the third side to form a triangle if the other sides are 8 and 7?

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78 What is the maximum length of the third side to form a triangle if the other sides are 8 and 7?

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Answer

third side < 8 + 7 third side < 15

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79 What is the minimum length of the third side to form a triangle if the other sides are 4 and 6?

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79 What is the minimum length of the third side to form a triangle if the other sides are 4 and 6?

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Answer

6 < third side + 4 2 < third side

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80 What is the minimum length of the third side to form a triangle if the other sides are 7 and 8?

Slide 134 (Answer) / 232

80 What is the minimum length of the third side to form a triangle if the other sides are 7 and 8?

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Answer

8 < third side + 7 1 < third side

Slide 135 / 232

Similar Triangles

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• f Contents

Slide 136 / 232

Recall that:

Congruence

Two objects are congruent if they can be moved, by any combination of translation, rotation and reflection, so that every part of each object overlaps. This is the symbol for congruence: If a is congruent to b, this would be shown as which is read as "a is congruent to b."

a b

Slide 137 / 232

Only line segments with the same length are congruent. Also, all congruent segments have the same length. We learned earlier that:

Congruent Line Segments

a b c d c d a b

Slide 138 / 232

Recall:

Congruent Angles

A B

∠ ∠

∠C ∠D Two angles are congruent if they have the same measure. Two angles are not congruent if they have different measures. A B C D If m∠A = m∠B If m∠C # m∠D

Slide 139 / 232 Congruent Triangles

Triangles are made up of three line segments AND three angles For one triangle to be congruent to another all three sides AND all three angles must be congruent.

Slide 140 / 232 Similar Triangles

If all the sides of two triangles are congruent, we will soon show that all the angles are also congruent. Therefore, the triangles are congruent. However, two triangles can have all their angles congruent, with all or none of their sides being congruent. In that case, they are said to be Similar Triangles.

Slide 141 / 232 Congruent Triangles

Congruent Triangles are also Similar Triangles since their angles are all congruent. Congruent triangles are therefore a special case of similar

• triangles. We will focus on similar triangles first, and then work

with congruent triangles in a later unit. Similar triangles represent a great tool to solve problems, and are the foundation of trigonometry.

Slide 142 / 232

Similar triangles have the same shape, but can have different sizes. If they have the same shape and are the same size, they are both similar and congruent. A B C D E F

Similar Triangles Have Proportional Sides Theorem Slide 143 / 232 Similar Triangles

This is the symbol for similarity So, the symbolic statement for Triangle ABC is similar to Triangle DEF is:

ΔABC ~ ΔDEF

Slide 144 / 232 Naming Similar Triangles

This statement tells you more than that the triangles are similar. It also tells you which angles are equal. In this case, that m∠A = m∠D m∠B = m∠E m∠C = m∠F And, thereby which are the corresponding, proportional, sides. AB corresponds to DE BC corresponds to EF CA corresponds to FD

ΔABC ~ ΔDEF

Slide 144 (Answer) / 232 Naming Similar Triangles

This statement tells you more than that the triangles are similar. It also tells you which angles are equal. In this case, that m∠A = m∠D m∠B = m∠E m∠C = m∠F And, thereby which are the corresponding, proportional, sides. AB corresponds to DE BC corresponds to EF CA corresponds to FD

ΔABC ~ ΔDEF

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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 145 / 232 Naming Similar Triangles

So, when you are naming similar triangles, the order of the letters matters. They don't have to be alphabetical. But they have to be named so that equal angles correspond to

• ne another.

ΔABC ~ ΔDEF

Slide 145 (Answer) / 232 Naming Similar Triangles

So, when you are naming similar triangles, the order of the letters matters. They don't have to be alphabetical. But they have to be named so that equal angles correspond to

• ne another.

ΔABC ~ ΔDEF

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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 146 / 232 Proving Triangles Similar

If you can prove that all three angles of two triangles are congruent, you have directly proven that they are similar. However, there are shortcuts to proving triangles similar. We will explore three sets of conditions that imply the three angles of two triangles are congruent, meaning that the triangles must be similar.

Slide 146 (Answer) / 232 Proving Triangles Similar

If you can prove that all three angles of two triangles are congruent, you have directly proven that they are similar. However, there are shortcuts to proving triangles similar. We will explore three sets of conditions that imply the three angles of two triangles are congruent, meaning that the triangles must be similar.

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Math Practice

The next few slides model various MP standards leading to the "shortcuts". All of the slides meet the MP4 & MP8 standards as well as the additional standards mentioned below: Slide #148 = proof: MP3 Slide #150-154 = construction: MP5 Slide #156-158 = explanation related to previous construction: MP3 & MP5

Slide 147 / 232 Angle-Angle Similarity Theorem

We know from the Triangle Sum Theorem that the sum of the interior angles of a triangle is always 180º. So, if two triangles have two pair of congruent angles which sum to x, then the third angle in both triangles must be (180 - x)º ....forming three congruent pairs of angles. One way to prove that two triangles are similar is to prove that two of the angles in each triangle are congruent.

Slide 148 / 232 Angle-Angle Similarity Theorem

If two angles of a triangle are congruent to two angles of another triangle, their third angles are congruent and the triangles are similar. Here's the proof: Statement Reason 1 ∠A and ∠B in ΔABC are ≅ to ∠D and ∠E in ΔDEF Given 2 m∠A = m∠D; m∠B = m∠E Definition of Congruent Angles 3 m∠A+ m∠B + m∠C = 180º m∠D+ m∠E + m∠F = 180º Triangle Sum Theorem 4 m∠C =180º - (m∠A + m∠B) m∠F =180º- (m∠D + m∠E) Subtraction Property of Equality 5 m∠C =180º - (m∠A + m∠B) m∠F =180º- (m∠A + m∠B) Substitution Property of Equality 6 m∠C = m∠F Substitution Property of Equality 7 ΔABC and ΔDEF are similar Definition of Similarity

Slide 149 / 232

If two triangles have their sides proportional, the triangles will be equiangular and will have those angles equal which their corresponding sides subtend. Euclid - Book Six: Proposition 5 Equiangular triangles are similar, so this states that triangles with proportional sides are similar. This is a second way to prove triangles are similar: If you can prove that all three pairs of sides in two triangles are proportional, then you have proven the triangles similar.

Side-Side-Side Similarity Theorem Slide 150 / 232 Side-Side-Side Similarity Theorem

This follows from the way we constructed congruent angles. We made use of the fact that if angles are congruent, their sides are separating at the same rate as you move away from the vertex. Here's the drawing we used to construct ∠ABC so it would be congruent to ∠FGH. F G H A C B

Slide 151 / 232 Side-Side-Side Similarity Theorem

If we draw the green line segments connecting the points where the blue arcs intersect the rays, we can see that the length of that segment would be the same for both angles. Since the angles are congruent, the line segment opposite those angles will also be congruent, if it intersects both sides of the angle at the same distance from the vertex in both cases. F G H A C B D E

Slide 152 / 232 Side-Side-Side Similarity Theorem

In this case the segments AC and DE will be congruent since segments GD and GE are also congruent to segments AB and BC. Therefore ΔDEG is congruent to ΔABC, since all the sides and angles are the same. Changing the scale of ΔABC won't change the angle measures. The sides would then be in proportion to those of ΔDEG, but not equal. F G H A C B D E

Slide 153 / 232

A C B

Side-Side-Side Similarity Theorem

The diagram below shows an expansion of ΔABC and we see that the measures of the angles are unchanged. They are still similar triangles. The corresponding sides are in proportion. F G H D E

Slide 154 / 232

A C B

Side-Side-Side Similarity Theorem

Removing the arcs and shifting the smaller triangle within the larger makes it clear that all angles are congruent and the sides are in proportion. So, the second way to prove triangles similar is to show that all their sides are in proportion. F D E G H

Slide 155 / 232 Side-Angle-Side Similarity Theorem

If two triangles have one angle equal to another and the sides about the equal angle are in proportion, the triangles will be equiangular and will have those angles equal which the corresponding sides subtend. Euclid's Elements - Book Six: Proposition 6 The third way to prove triangles are similar is to show they share an angle which is equal and the two sides forming that angle are proportional in the two triangles.

Slide 156 / 232 Side-Angle-Side Similarity Theorem

This directly follows from the work we just did to show that Side-Side-Side proportionality can be used to prove triangles are similar. If you recall, the line segment which makes up the third side of a triangle is completely defined by its opposite angle and the lengths of the other two sides.

Slide 157 / 232 Side-Angle-Side Similarity Theorem

If the angles are congruent and the two sides of the angle are in proportion, the third side must also be in proportion. If all three sides are in proportion, the triangles must be similar due to the Side-Side-Side Theorem. You can see that on the next page.

Slide 158 / 232

A B C D E F

Side-Angle Side Similarity Theorem

If ∠B ≅ ∠E and segments AB and BC are proportional to segments ED and EF, then segment AC must also be proportional to segment DF. Since all the sides are in proportion, the triangles are similar.

Slide 159 / 232 Common Error

You CANNOT prove triangles similar using Side-Side-Angle. This is not the same as Side-Angle-Side. As shown below, two triangles can have two corresponding sides and one corresponding angle congruent, but NOT be similar.

Slide 160 / 232

81 Which theorem allows you to prove these two triangles are similar? A Angle-Angle B Side-Angle-Side C Side-Side-Side D They may not be similar x x E They are not similar

Slide 160 (Answer) / 232

81 Which theorem allows you to prove these two triangles are similar? A Angle-Angle B Side-Angle-Side C Side-Side-Side D They may not be similar x x E They are not similar

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Answer

A

x x

Slide 161 / 232

82 Which theorem allows you to prove these two triangles are similar? A Angle-Angle B Side-Angle-Side C Side-Side-Side D They may not be similar E They are not similar

Slide 161 (Answer) / 232

82 Which theorem allows you to prove these two triangles are similar? A Angle-Angle B Side-Angle-Side C Side-Side-Side D They may not be similar E They are not similar

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Answer

D; not enough information

Slide 162 / 232

83 Which theorem allows you to prove these two triangles are similar? A Angle-Angle B Side-Angle-Side C Side-Side-Side D They may not be similar 6 4 8 8 12 E They are not similar 16

Slide 162 (Answer) / 232

83 Which theorem allows you to prove these two triangles are similar? A Angle-Angle B Side-Angle-Side C Side-Side-Side D They may not be similar 6 4 8 8 12 E They are not similar 16

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Answer

C Slide 163 / 232

84 Which theorem allows you to prove these two triangles are similar? A Angle-Angle B Side-Angle-Side C Side-Side-Side D They may not be similar 4 8 3 6 6 10 E They are not similar

Slide 163 (Answer) / 232

84 Which theorem allows you to prove these two triangles are similar? A Angle-Angle B Side-Angle-Side C Side-Side-Side D They may not be similar 4 8 3 6 6 10 E They are not similar

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Answer

E; the ratios are not the same

Slide 164 / 232

85 Which theorem allows you to prove these two triangles are similar? 4 8 3 6 x x A Angle-Angle B Side-Angle-Side C Side-Side-Side D They may not be similar E They are not similar

Slide 164 (Answer) / 232

85 Which theorem allows you to prove these two triangles are similar? 4 8 3 6 x x A Angle-Angle B Side-Angle-Side C Side-Side-Side D They may not be similar E They are not similar

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Answer

B Slide 165 / 232

86 Which theorem allows you to prove these two triangles are similar? A Angle-Angle B Side-Angle-Side C Side-Side-Side D They may not be similar 4 3 x 8 6 x E They are not similar

Slide 165 (Answer) / 232

86 Which theorem allows you to prove these two triangles are similar? A Angle-Angle B Side-Angle-Side C Side-Side-Side D They may not be similar 4 3 x 8 6 x E They are not similar

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Answer

D

4 3 x 8 6 x These are NOT similar 4 3 x 8 6 x These are similar x x

Slide 166 / 232

87 Which theorem allows you to prove these two triangles are similar? A Angle-Angle B Side-Angle-Side C Side-Side-Side D They may not be similar E They are not similar

Slide 166 (Answer) / 232

87 Which theorem allows you to prove these two triangles are similar? A Angle-Angle B Side-Angle-Side C Side-Side-Side D They may not be similar E They are not similar

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Answer

C Slide 167 / 232

88 Which theorem allows you to prove these two triangles are similar? A Angle-Angle B Side-Angle-Side C Side-Side-Side D They may not be similar E They are not similar

Slide 167 (Answer) / 232

88 Which theorem allows you to prove these two triangles are similar? A Angle-Angle B Side-Angle-Side C Side-Side-Side D They may not be similar E They are not similar

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Answer

D

x x x y These are similar These are NOT similar

Slide 168 / 232

89 Which theorem allows you to prove these two triangles are similar? A Angle-Angle B Side-Angle-Side C Side-Side-Side D They may not be similar E They are not similar

Slide 168 (Answer) / 232

89 Which theorem allows you to prove these two triangles are similar? A Angle-Angle B Side-Angle-Side C Side-Side-Side D They may not be similar E They are not similar

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Answer

B Slide 169 / 232

90 Which theorem allows you to prove these two triangles are similar? A Angle-Angle B Side-Angle-Side C Side-Side-Side D They may not be similar E They are not similar A B C D E Note that BC is parallel to DE.

Slide 169 (Answer) / 232

90 Which theorem allows you to prove these two triangles are similar? A Angle-Angle B Side-Angle-Side C Side-Side-Side D They may not be similar E They are not similar A B C D E Note that BC is parallel to DE.

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Answer

A; using any 2

• f the 3

angles marked in diagram

A B C D E

Slide 170 / 232

Similar triangles have the same shape, but can have different

• sizes. If they have the same shape and are the same size,

they are congruent. If they have the same shape and are different sizes, they are similar and their sides are in proportion. A B C D E F

Similar Triangles Have Proportional Sides Theorem Slide 171 / 232

If two triangles are similar, all of their corresponding sides are in proportion. *While Euclid does prove this theorem, his proof relies on

• ther theorems which would have to be proven first and would

take us beyond the scope of this course. So, we'll just rely on this theorem and note that the proof is available in The Elements by Euclid - Book Six: Proposition 5.

Similar Triangles Have Proportional Sides Theorem Slide 172 / 232 Similar Triangles and Proportionality

A B C D E F

In the triangles below, if we know that m∠A = m∠D, m∠B = m∠E, and m∠C = m∠F, then we know that the triangles are similar.

Slide 173 / 232 Similar Triangles and Proportionality

A B C D E F

We also then know that the corresponding sides are proportional. The symbol for proportional is the Greek letter, alpha: # AB α DE, since AB corresponds to DE BC α EF, since BC corresponds to EF AC α DF, since AC corresponds to DF

Slide 174 / 232 Corresponding Sides

A B C D E F

Our work with similar triangles and our future work with congruent triangles requires us to identify the corresponding sides. One way to do that is to locate the sides opposite congruent

• angles. If we know triangles ABC and EDF are similar and that

angle A is congruent to angle D, then the sides opposite A and D are in proportion: BC α EF

Slide 175 / 232 Corresponding Sides

A B C D E F

Another way of identifying corresponding sides is to use Euclid's description "...those angles [are] equal which their corresponding sides subtend." Below, since angle A is equal to angle D and angle B is equal to angle E, then sides AB and DE are in proportion.

Slide 176 / 232 Corresponding Sides

A B C D E F

Either approach works; use the one you find easiest. Identify corresponding sides as the sides connecting equal angles

• r the sides opposite equal angles...you'll get the same result.

Slide 177 / 232 Similar Triangles and Proportionality

A B C D E F

Another way of saying two sides are proportional is to say that

• ne is a scaled-up version of the other. If you multiply all the

sides of one triangle by the same scale factor, k, you get the

• ther triangle. In this case, if ΔABC is k times as big as ΔDEF,

then: AB = kDE BC = kEF AC = kDF

Slide 178 / 232 Similar Triangles and Proportionality

A B C D E F

Or, dividing the corresponding sides yields: AB BC AC DE EF DF = k = = This property of proportionality is very useful in solving problems using similar triangles, and provides the foundation for trigonometry.

Slide 179 / 232

91 If m∠A = m∠D, m∠B = m∠E, and m∠C = m∠F, identify which side corresponds to side AB. A DE B EF C FG A B C D E F

Slide 179 (Answer) / 232

91 If m∠A = m∠D, m∠B = m∠E, and m∠C = m∠F, identify which side corresponds to side AB. A DE B EF C FG A B C D E F

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Answer

A Slide 180 / 232

92 If m∠I = m∠M, m∠H = m∠N, and m∠J = m∠L, identify which side corresponds to side IJ. A MN B NL C ML I J H M N L

Slide 180 (Answer) / 232

92 If m∠I = m∠M, m∠H = m∠N, and m∠J = m∠L, identify which side corresponds to side IJ. A MN B NL C ML I J H M N L

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Answer

C Slide 181 / 232

A B C 8 D E F 4

Example - Proportional Sides

Given that ΔABC is similar to ΔDEF, and given the indicated lengths, find the lengths AB and BC. 5 7

Slide 182 / 232 Example - Proportional Sides

Since the triangles are similar we know that the following relationship holds between all the corresponding sides. First, let's find the constant of proportionality, k, by using the two sides for which we have values: AC and DF. What ratio could I write to determine the value of k? AB BC AC ED EF DF = k = =

A B C 8 D E F 4 5 7

Slide 182 (Answer) / 232 Example - Proportional Sides

Since the triangles are similar we know that the following relationship holds between all the corresponding sides. First, let's find the constant of proportionality, k, by using the two sides for which we have values: AC and DF. What ratio could I write to determine the value of k? AB BC AC ED EF DF = k = =

A B C 8 D E F 4 5 7

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Math Practice

The last question addresses MP2

Slide 183 / 232

A B C 5 7 8 D E F 4

Example - Proportional Sides

AB BC AC ED EF DF = k = 2 = = AC 8 DF 4 = = k = 2 That means that the other two sides of ΔABC will also be twice as large as the corresponding sides of ΔDEF How can we write the proportions required to calculate AB and BC?

Slide 183 (Answer) / 232

A B C 5 7 8 D E F 4

Example - Proportional Sides

AB BC AC ED EF DF = k = 2 = = AC 8 DF 4 = = k = 2 That means that the other two sides of ΔABC will also be twice as large as the corresponding sides of ΔDEF How can we write the proportions required to calculate AB and BC?

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Math Practice

The last question addresses MP2

Slide 184 / 232

A B C

5 7 8

D E F 4

Example - Proportional Sides

AB ED = 2 BC EF = 2 AB 5 = 2 AB = 10 BC 7 = 2 BC = 14

Slide 185 / 232

93 Given that m∠A = m∠D, m∠B = m∠E, and m∠C = m∠F. If BC = 8, DE = 6, and AB = 4, EF = ? A B C D E F

Slide 185 (Answer) / 232

93 Given that m∠A = m∠D, m∠B = m∠E, and m∠C = m∠F. If BC = 8, DE = 6, and AB = 4, EF = ? A B C D E F

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Answer

4 6 8 x = 4x = 48 4 4 x = 12 = EF

Slide 186 / 232

94 Given that ΔJIH is similar to ΔLMN; find the length of LM. I J H M N L 14 10 12 5

Slide 186 (Answer) / 232

94 Given that ΔJIH is similar to ΔLMN; find the length of LM. I J H M N L 14 10 12 5

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Answer

12 x 10 5 = 60 = 10x 10 10 x = 6 = LM

Slide 187 / 232

95 Given that ΔJIH is similar to ΔLMN; find the length of LN. I J H M N L 14 10 12 5

Slide 187 (Answer) / 232

95 Given that ΔJIH is similar to ΔLMN; find the length of LN. I J H M N L 14 10 12 5

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Answer

14 x 10 5 = 70 = 10x 10 10 x = 7 = LN

Slide 188 / 232

96 Given that BC is parallel to DE and the given lengths, find the length of DE. A B C D E 8 6 4

Slide 188 (Answer) / 232

96 Given that BC is parallel to DE and the given lengths, find the length of DE. A B C D E 8 6 4

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Answer

8 12 6 x = 8x = 72 8 8 x = 9 = DE

Slide 189 / 232

97 Given that BC is parallel to DE and the given lengths, find the length of DB. A B C D E 9 7 3

Slide 189 (Answer) / 232

97 Given that BC is parallel to DE and the given lengths, find the length of DB. A B C D E 9 7 3

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Answer

3 9 7 x = 3x = 63 3 3 x = 21 = AD BD + 7 = 21 BD = 14

Slide 190 / 232 Example - Similarity & Proportional Sides

D P K 12 9 18 R L B 6 12 10

Determine if the triangles are similar. If they are similar write a similarity statement. If they are not similar, explain why.

Slide 191 / 232 Example - Similarity & Proportional Sides

D P K 12 9 18 R L B 6 12 10

To identify the corresponding sides without wasting a lot of time, first list all the sides from shortest to longest of both triangles and compare to see if they are all proportional. Then you can identify corresponding sides and the constant of proportionality.

Slide 191 (Answer) / 232 Example - Similarity & Proportional Sides

D P K 12 9 18 R L B 6 12 10

To identify the corresponding sides without wasting a lot of time, first list all the sides from shortest to longest of both triangles and compare to see if they are all proportional. Then you can identify corresponding sides and the constant of proportionality.

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Math Practice

This example addresses MP4 & MP6.

Slide 192 / 232 Example - Similarity & Proportional Sides

D P K 15 9 18 R L B 6 12 10 Side of ΔPDK Length Side of ΔBRL Length Ratio DK 9 BR 6 1.5 PD 15 RL 10 1.5 PK 18 BL 12 1.5 All corresponding sides are in the ratio of 1.5:1, so the triangles are

• similar. This also provides the order of the sides, so we can say that

ΔKDP is similar to ΔBRL. Check to make sure that all the sides are in the correct order.

Slide 193 / 232

98 If these triangles are similar, enter the constant of proportionality, k, between the larger and smaller

• triangle. If they are not, enter zero.

D P K 12 9 18 R L B 6 12 10

Slide 193 (Answer) / 232

98 If these triangles are similar, enter the constant of proportionality, k, between the larger and smaller

• triangle. If they are not, enter zero.

D P K 12 9 18 R L B 6 12 10

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Answer

DK = 9 PD = 12 PK = 18 BR = 6 RL = 10 BL = 12 9 6 12 5 ≠ Not similar; 0

Slide 194 / 232

99 If these triangles are similar, enter the constant of proportionality, k, between the larger and smaller

• triangle. If they are not, enter zero.

52° 1 2 3 R S T 52° 2 4 6 X Y Z

Slide 194 (Answer) / 232

99 If these triangles are similar, enter the constant of proportionality, k, between the larger and smaller

• triangle. If they are not, enter zero.

52° 1 2 3 R S T 52° 2 4 6 X Y Z

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Answer

RS = 1 RT = 2 ST = 3 YZ = 2 YX = 4 ZX = 6 1 2 2 4 3 6 = =

1/2 = 0.5 = k

Slide 195 / 232

100 If these triangles are similar, enter the constant of proportionality, k, between the larger and smaller

• triangle. If they are not, enter zero.

P R S 3 4.2 6 B C D 2 2.8 4

Slide 195 (Answer) / 232

100 If these triangles are similar, enter the constant of proportionality, k, between the larger and smaller

• triangle. If they are not, enter zero.

P R S 3 4.2 6 B C D 2 2.8 4

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Answer

PR = 3 RS = 4.2 PS = 6 BC = 2 CD = 2.8 BD = 4 3 2 4.2 2.8 6 4 = =

3/2 = 1.5 = k

Slide 196 / 232

A B C D E

Side Splitter Theorem

Any line parallel to a side of a triangle will form a triangle which is similar to the first triangle. It also makes all the sides proportional, splitting them...hence the name of the theorem. We're going to start off with the first part of the Side Splitter Theorem Proof, proving the triangles to be similar.

Slide 197 / 232

A B C D E

Proof of Side Splitter Theorem

Given: BC is parallel to DE Prove: ΔABC ~ ΔADE.

Slide 198 / 232

101 What is the reason for step 2? A Angle-Angle Similarity Theorem B Side-Side-Side Similarity Theorem C Reflexive Property of Congruence D When two parallel lines are intersected by a transversal, the corresponding angles are congruent. E When two parallel lines are intersected by a transversal, the alternate interior angles are congruent.

A B C D E

Answer

Statement Reason 1 BC is parallel to DE Given 2 ∠ABC ≅ ∠D; ∠ACB ≅ ∠E ? 3 ∠A ≅ ∠A ? 4 ΔABC ~ ΔADE ?

Slide 199 / 232

102 What is the reason for step 3? A Angle-Angle Similarity Theorem B Side-Side-Side Similarity Theorem C Reflexive Property of Congruence D When two parallel lines are intersected by a transversal, the corresponding angles are congruent. E When two parallel lines are intersected by a transversal, the alternate interior angles are congruent.

A B C D E

Statement Reason 1 BC is parallel to DE Given 2 ∠ABC ≅ ∠D; ∠ACB ≅ ∠E ? 3 ∠A ≅ ∠A ? 4 ΔABC ~ ΔADE ?

Answer

Slide 200 / 232

103 What is the reason for step 4? A Angle-Angle Similarity Theorem B Side-Side-Side Similarity Theorem C Reflexive Property of Congruence D When two parallel lines are intersected by a transversal, the corresponding angles are congruent. E When two parallel lines are intersected by a transversal, the alternate interior angles are congruent. A B C D E

Statement Reason 1 BC is parallel to DE Given 2 ∠ABC ≅ ∠D; ∠ACB ≅ ∠E ? 3 ∠A ≅ ∠A ? 4 ΔABC ~ ΔADE ?

Answer

Slide 201 / 232 Proof of Side Splitter Theorem

Given: BC is parallel to DE Prove: ΔABC ~ ΔADE A B C D E Statement Reason 1 BC is parallel to DE Given 2 ∠ABC ≅ ∠D; ∠ACB ≅ ∠E When two parallel lines are intersected by a transvesal, the corresponding angles are congruent 3 ∠A ≅ ∠A Reflexive Property of Congruence 4 ΔABC ~ ΔADE Angle-Angle Similarity Theorem

Slide 202 / 232 Proof of Side Splitter Theorem

Now, we know that ΔABC ~ ΔADE. The remaining steps of this proof use the properties of proportional triangles and equality. A B C D E BD CE AB AC = Given: ΔABC ~ ΔADE Prove:

Slide 203 / 232

Statement Reason 1 ΔABC ~ ΔADE Given 2 AD AE AB AC ? 3 AB + BD = AD, AC + CE = AE ? 4 AB + BD AC + CE AB AC ? 5 AB BD AC CE AB AB AC AC Addition Property of Fractions 3 2 3 + 2 5 7 7 7 7 6 BD CE AB AC ? 7 BD CE AB AC ? = = = + + 1 + = 1 + =

Proof of Side Splitter Theorem

The table below will be used to answer the next 5 Response Questions + = = e.g.

( )

Slide 204 / 232

104 What is the reason for step 2? A Segment Addition Postulate B Subtraction Property of Equality C Substitution Property of Equality D When two triangles are similar, their corresponding sides are proportional. E Simplifying the Equation

A B C D E

Slide 204 (Answer) / 232

104 What is the reason for step 2? A Segment Addition Postulate B Subtraction Property of Equality C Substitution Property of Equality D When two triangles are similar, their corresponding sides are proportional. E Simplifying the Equation

A B C D E

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Answer

D Slide 205 / 232

105 What is the reason for step 3? A Segment Addition Postulate B Subtraction Property of Equality C Substitution Property of Equality D When two triangles are similar, their corresponding sides are proportional. E Simplifying the Equation

A B C D E

Slide 205 (Answer) / 232

105 What is the reason for step 3? A Segment Addition Postulate B Subtraction Property of Equality C Substitution Property of Equality D When two triangles are similar, their corresponding sides are proportional. E Simplifying the Equation

A B C D E

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Answer

A

Slide 206 / 232

106 What is the reason for step 4? A Segment Addition Postulate B Subtraction Property of Equality C Substitution Property of Equality D When two triangles are similar, their corresponding sides are proportional. E Simplifying the Equation

A B C D E

Slide 206 (Answer) / 232

106 What is the reason for step 4? A Segment Addition Postulate B Subtraction Property of Equality C Substitution Property of Equality D When two triangles are similar, their corresponding sides are proportional. E Simplifying the Equation

A B C D E

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Answer

C Slide 207 / 232

107 What is the reason for step 6? A Segment Addition Postulate B Subtraction Property of Equality C Substitution Property of Equality D When two triangles are similar, their corresponding sides are proportional. E Simplifying the Equation

A B C D E

Slide 207 (Answer) / 232

107 What is the reason for step 6? A Segment Addition Postulate B Subtraction Property of Equality C Substitution Property of Equality D When two triangles are similar, their corresponding sides are proportional. E Simplifying the Equation

A B C D E

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Answer

E Slide 208 / 232

108 What is the reason for step 7? A Segment Addition Postulate B Subtraction Property of Equality C Substitution Property of Equality D When two triangles are similar, their corresponding sides are proportional. E Simplifying the Equation

A B C D E

Slide 208 (Answer) / 232

108 What is the reason for step 7? A Segment Addition Postulate B Subtraction Property of Equality C Substitution Property of Equality D When two triangles are similar, their corresponding sides are proportional. E Simplifying the Equation

A B C D E

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Answer

B

Slide 209 / 232

Statement Reason 1 ΔABC ~ ΔADE Given 2 AD AE AB AC When two triangles are similar, their corresponding sides are proportional 3 AB + BD = AD, AC + CE = AE Segment Addition Postulate 4 AB + BD AC + CE AB AC Substitution Property of Equality 5 AB BD AC CE AB AB AC AC Addition Property of Fractions 3 2 3 + 2 5 7 7 7 7 6 BD CE AB AC Simplify the Equation 7 BD CE AB AC Subtraction Property of Equality = = = + + 1 + = 1 + =

Proof of Side Splitter Theorem

+ = = e.g.

( )

BD CE AB AC = Given: ΔABC ~ ΔADE Prove:

Slide 210 / 232

A B C D E

Converse of Side Splitter Theorem

If a line divides the two sides of a triangle proportionally, then the line is parallel to the third side. A proof of this theorem is one of the homework problems that you will work on tonight.

Slide 211 / 232

109 Find the value of x to prove that AB is parallel to ER. 27 x 18 12 R E A B D

Slide 211 (Answer) / 232

109 Find the value of x to prove that AB is parallel to ER. 27 x 18 12 R E A B D

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Answer

27 18 x 12 = 324 = 18x 18 18 x = 18

Slide 212 / 232

110 Find the value of x to prove that FC is parallel to MN. J M N C F x 9 6 8

Slide 212 (Answer) / 232

110 Find the value of x to prove that FC is parallel to MN. J M N C F x 9 6 8

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Answer

8 6 9 x = 8x = 54x 8 8 x = 6.75

Slide 213 / 232

111 Find the value of y. 6 10 12 y

Slide 213 (Answer) / 232

111 Find the value of y. 6 10 12 y

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Answer

12 y 10 6 = 72 = 10y 10 10 y = 7.2

Slide 214 / 232

112 Find the value of y. 4 14 12 y

Slide 214 (Answer) / 232

112 Find the value of y. 4 14 12 y

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Answer

14 y 12 4 = 56 = 12y 12 12 x = 4.6

Slide 215 / 232

113 Find the value of y. 24 15 y 6

Slide 215 (Answer) / 232

113 Find the value of y. 24 15 y 6

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Answer

21 24 15 y = 360 = 21y 21 21 x 17.1

Slide 216 / 232

PARCC Sample Question and Applications

Return to Table

• f Contents

Slide 217 / 232

114 The figure ΔABC ~ ΔDEF with side lengths as indicated. What is the value of x? F D E 9 5 7 C B A 27 21 x

From PARCC EOY sample test

Slide 217 (Answer) / 232

114 The figure ΔABC ~ ΔDEF with side lengths as indicated. What is the value of x? F D E 9 5 7 C B A 27 21 x

From PARCC EOY sample test

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Answer

5 x 9 27 = 9x = 135 5 x 7 21 = 7x = 105 OR x = 15

Slide 218 / 232

How can we use similar figures to solve real-life problems? Using similar triangles and indirect measurement, we can find large distances and the heights of trees, flagpoles, and buildings. What is the difference between direct measurement and indirect measurement?

Using Similar Triangles Slide 218 (Answer) / 232

How can we use similar figures to solve real-life problems? Using similar triangles and indirect measurement, we can find large distances and the heights of trees, flagpoles, and buildings. What is the difference between direct measurement and indirect measurement?

Using Similar Triangles

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Answer

When two figures are similar, you can use proportions to find unknown side lengths. Indirect measurement uses proportions to find measurements when direct measurement is not possible. Direct measurement is when an object can be directly measured, meaning that you can actually pick it up and measure the sides. Q's on this slide address MP1 & MP6

Slide 219 / 232

How can we find the distance across the Grand Canyon?

Grand Canyon National Park, AZ

Using Similar Triangles

Slide 219 (Answer) / 232

How can we find the distance across the Grand Canyon?

Grand Canyon National Park, AZ

Using Similar Triangles

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Math Practice

This example (this slide and the next 5) address MP1, MP2, MP3, MP4 & MP5

Slide 220 / 232

First, construct right triangle ΔABC.

• 1. Identify a landmark at point A.
• 2. Place a marker at point B directly across from point A.
• 3. Walk to point C, place a marker and measure the distance of BC.

Using Similar Triangles Slide 221 / 232

Then, construct right triangle ΔEDC.

• 1. Walk to point D, place a marker and measure the distance of CD.
• 2. Walk to point E, place a marker and measure the distance of DE.

Using Similar Triangles Slide 222 / 232

How can you prove that ΔABC ~ ΔEDC? How can you find the distance across the Grand Canyon?

Using Similar Triangles Slide 223 / 232

ΔABC ~ ΔEDC Why? Why? Why? ∠DCE ≅ ∠BCA ∠CDE ≅ ∠CBA

Using Similar Triangles Slide 223 (Answer) / 232

ΔABC ~ ΔEDC Why? Why? Why? ∠DCE ≅ ∠BCA ∠CDE ≅ ∠CBA

Using Similar Triangles

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Answer

∠DCE≅∠BCA Vertical angles are congruent ∠CDE≅∠CBA All right angles are congruent ΔABC~ ΔEDC By AA~

Slide 224 / 232

How do you find d? Write a statement of proportionality that uses d.

Using Similar Triangles

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Slide 224 (Answer) / 232

How do you find d? Write a statement of proportionality that uses d.

Using Similar Triangles

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Answer

The distance across the Grand Canyon is 200 ft. AB ED BC DC = d 250 400 500 = 500d = 100,000 d = 200

Slide 225 / 232

How can we find the height of the Washington Monument when there are no shadows?

Using Similar Triangles Slide 226 / 232

We are going to use a mirror trick to find the height of the Washington

• Monument. This is another method of indirect measurement.

Place a mirror with cross hairs (an X) drawn on it flat on the ground between yourself and the Washington Monument. Look into the mirror and walk to a point at which you see the top of the Washington Monument lining up with the mirror's cross hairs. The light rays from the top of the Washington Monument to the mirror and back up to your eye form equal angles.

Using Similar Triangles Slide 226 (Answer) / 232

We are going to use a mirror trick to find the height of the Washington

• Monument. This is another method of indirect measurement.

Place a mirror with cross hairs (an X) drawn on it flat on the ground between yourself and the Washington Monument. Look into the mirror and walk to a point at which you see the top of the Washington Monument lining up with the mirror's cross hairs. The light rays from the top of the Washington Monument to the mirror and back up to your eye form equal angles.

Using Similar Triangles

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Math Practice

This example (this slide and the next 4 slides) addresses MP1, MP2, MP3, MP4, MP5 & MP7

Slide 227 / 232

In Physics,

Using Similar Triangles

angle of incidence angle of reflection reflected ray incident ray surface angle of reflection = angle of incidence

Slide 228 / 232

Measure the distance from you to the mirror and the Washington Monument to the mirror. How can you prove that ΔABC ~ ΔDEF? How can you find the height of the Washington Monument?

Using Similar Triangles Slide 229 / 232 Using Similar Triangles

Why? Why? Why? ΔABC ~ ΔADE ∠CAB ≅∠EAD ∠ACB ≅∠EAD

Slide 229 (Answer) / 232 Using Similar Triangles

Why? Why? Why? ΔABC ~ ΔADE ∠CAB ≅∠EAD ∠ACB ≅∠EAD

[This object is a pull tab] ∠CAB ≅∠EAD angle of reflection = angle of incidence ∠ACB ≅∠EAD All right angles are congruent ΔABC ~ ΔADE By AA~

Slide 230 / 232

How do you find h? Write a statement of proportionality that uses h.

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Using Similar Triangles Slide 230 (Answer) / 232

How do you find h? Write a statement of proportionality that uses h.

click

Using Similar Triangles

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Answer

The height of the Washington Monument is 555 ft. BC DE AC AE = h 68 12,000 122.5 = 122.5 = 816,000 h = 6,661.22 in. h ≈ 555 ft

Slide 231 / 232

115 Your little sister wants to know the height of the giraffe. You place a mirror on the ground and stand where you can see the top of the giraffe as shown. How tall is the giraffe? A 189 in B 21 ft C 15.75 ft D 18.9 ft You 5 ft 3 in g 15 ft 5 ft

Slide 231 (Answer) / 232

115 Your little sister wants to know the height of the giraffe. You place a mirror on the ground and stand where you can see the top of the giraffe as shown. How tall is the giraffe? A 189 in B 21 ft C 15.75 ft D 18.9 ft You 5 ft 3 in g 15 ft 5 ft

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Answer

C Slide 232 / 232

116 To find the width of a river, you use a surveying technique as shown. Set up the proportion to find the distance across the river. A B C D = 9 63 w 12 = 9 63 w 12 = 9 63 w 12 = 9 63 w 12

Slide 232 (Answer) / 232

116 To find the width of a river, you use a surveying technique as shown. Set up the proportion to find the distance across the river. A B C D = 9 63 w 12 = 9 63 w 12 = 9 63 w 12 = 9 63 w 12

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Answer

D