Geometry Triangles Triangles Return to Table of Contents - - PDF document

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Table of Contents

· Questions from Released PARCC Test · Triangle Sum Theorem · Exterior Angle Theorem · Triangles · Inequalities in Triangles · Similar Triangles

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Geometry Triangles

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Triangles

Slide 5 / 210 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 6 / 210 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 7 / 210 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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2 The letter on this triangle that represents a vertex is: A B C

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

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

Slide 19 / 210 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 two sides with equal length. The angles opposite those equal sides are of equal measure. x x

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 x 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. x x

Classifying Triangles Slide 26 / 210

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 triangle.

x x x

Classifying Triangles Slide 28 / 210 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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11 An obtuse triangle is _______________ an isosceles triangle. A Sometimes B Always C Never

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

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

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

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15 An equilateral triangle is also an isosceles triangle True False

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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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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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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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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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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

scalene, obtuse 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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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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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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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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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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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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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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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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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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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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Triangle Sum Theorem

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Slide 52 / 210 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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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

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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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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 56 / 210

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

Triangle Sum Theorem Slide 57 / 210

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 58 / 210

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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Example: Triangle Sum Theorem

320 J K L 200 Find the measure of the missing angle.

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30 What is m∠B? A B C 52° 53°

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

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

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

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Solve for x

55° (12x+8)° (8x-3)° P Q R

Example Slide 65 / 210

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

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35 What is the measure of ∠B? C B A

(3x-17)0 (x+40)0 (2x-5)0

Slide 67 / 210 Corollary to Triangle Sum Theorem

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

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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 69 / 210

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

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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

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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 Answer 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

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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

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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

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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

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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 77 / 210 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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43 In a right triangle, the two acute angles sum to 90° True False

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

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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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46 Solve for x

What are the measures of the three angles?

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47 m∠1 + m∠2 = 1 2 3

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48 m∠1 + m∠3 = 1 2 3

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

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Exterior Angle Theorem

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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 x

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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 x

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

Exterior Angles Slide 89 / 210

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

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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 x x

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

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A B C x 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 Slide 94 / 210

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 x D

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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 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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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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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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Example: Using the Exterior Angle Theorem

140º Xº Xº P Q R What is the value of x?

Slide 104 / 210 Example

Solve for x and y. 21° 34° x° y°

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xº yº 75º 50º

Example

Solve for x and y.

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60 Solve for x. xº yº 60º 55º

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61 Solve for y. xº yº 60º 55º

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

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

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

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

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Example

Find the missing angles in the diagram. 60° 7 103° 43° 45° 30° 5 4 3 2 1

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

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67 Find the measure of ∠2. 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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69 Find the measure of ∠4.

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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Inequalities in Triangles

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Slide 119 / 210 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 / 210 Angle Inequalities in a Triangle

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

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

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

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

Slide 127 / 210 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 128 / 210 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 / 210 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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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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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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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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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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Similar Triangles

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Slide 135 / 210

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

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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

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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 138 / 210 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 139 / 210 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 140 / 210 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.

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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 142 / 210 Similar Triangles

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

Slide 143 / 210 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 DEF DEF ΔABC Δ

Slide 144 / 210 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.

DEF DEF ΔABC Δ

Slide 145 / 210 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 / 210 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 (180o - x) ....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 147 / 210

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 148 / 210

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 149 / 210 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 150 / 210 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 151 / 210 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 152 / 210

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 153 / 210

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 154 / 210 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 155 / 210 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 156 / 210 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 157 / 210

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 158 / 210 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 159 / 210

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 / 210

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 / 210

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 / 210

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 / 210

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 / 210

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 / 210

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 / 210

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 / 210

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 / 210

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 / 210

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. As we will learn later, it also makes all the sides proportional, splitting them...hence the name of the theorem.

Slide 170 / 210

A B C D E

Proof of Side Splitter Theorem

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

Slide 171 / 210

91 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 172 / 210

92 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.

Answer

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 ?

Slide 173 / 210

93 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 174 / 210

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 175 / 210

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 176 / 210

The converse is also true, and will prove very useful. 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 177 / 210 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 178 / 210 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 179 / 210 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 180 / 210 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 181 / 210 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 182 / 210 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 183 / 210 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 184 / 210

94 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 185 / 210

95 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 186 / 210

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 187 / 210 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 AB BC AC ED EF DF = k = =

A B C

8

D E F 4

5 7

Slide 188 / 210

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

Slide 189 / 210

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 190 / 210

96 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 191 / 210

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

Slide 192 / 210

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

Slide 193 / 210

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

Slide 194 / 210

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

Slide 195 / 210 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 196 / 210 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 197 / 210

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 198 / 210

101 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 199 / 210

102 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 200 / 210

103 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 201 / 210

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.

Slide 202 / 210

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

Slide 203 / 210

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

Slide 204 / 210

106 Find the value of y. 6 10 12 y

Slide 205 / 210

107 Find the value of y. 4 14 12 y

Slide 206 / 210

108 Find the value of y. 24 15 y 6

Slide 207 / 210

Questions from Released PARCC Test

Return to Table

• f Contents

Slide 208 / 210

Question 1/7

Slide 209 / 210

109 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

Slide 210 / 210