Geometry Triangles

Transcription

1 1

2 Geometry Triangles

3 Table of Contents Click on the topic to go to that section Triangles Triangle Sum Theorem Exterior Angle Theorem Inequalities in Triangles Similar Triangles PARCC Sample Question and Applications 3

4 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 others. 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. Math Practice 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. 4

5 Triangles Return to Table of Contents 5

6 Geometric Figures Definition 13. A boundary is that which is an extremity of anything. Definition 14. A figure is that which is contained by any boundary or boundaries. 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. 6

7 Geometric Figures 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. 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. 7

8 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. 8

9 1 The letter on this triangle that corresponds to a side is: A C B 9

10 2 The letter on this triangle that represents a vertex is: C B A 10

11 Parts of a Triangle Each vertex is named with a letter. C 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. A Name the 3 sides of this triangle B 11

12 3 What is the name of the side shown in red? A B C AB BC AC C A B 12

13 4 What is the name of the side shown in red? A B AB BC C C AC A B 13

14 5 Which of the following are names of this triangle? A ΔABC D ΔCAB B ΔBCA E all of these C ΔACB C A B 14

15 Parts of a Triangle C A side is opposite an angle if it does not touch it. Otherwise, it is adjacent to the angle. A B In the above, the red side is A, while the green sides are to A. 15

16 6 Which side is opposite angle B? A B C D AB CA BC None C A B 16

17 7 Which side is opposite angle A? A B AB CA C C D BC None A B 17

18 8 Which sides are adjacent to angle C? C A AB & BC B CA & BA C D BC & CA None A B 18

19 9 Which sides are adjacent to angle B? A B C D AB & BC CA & BA BC & CA None C A B 19

20 Types of Triangles In general, a triangle can have sides of all different lengths and angles of 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. 20

21 Classifying Triangles Definition 20: Of trilateral figures, an equilateral triangle is that which has its three sides equal, an isosceles triangle is that which has two of its sides alone equal, and a scalene triangle is that which has its three sides unequal 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. 21

22 Classifying Triangles 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. We will draw from both definitions, since in several cases both definitions apply to the same triangle. 22

23 Classifying Triangles Acute Triangles In an acute triangle, every angle of a triangle is acute. Definition 21: "...an acute-angled triangle is that which has its three angles acute." Notice that no angle is equal to or greater than 90º in this triangle. 23

24 Classifying Triangles Right Triangles A right triangle has one right angle and two acute angles. Definition 21: "...a right-angled triangle is that which has a right angle..." 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 other two sides are called the legs. 24

25 Classifying Triangles Isosceles Triangles An isosceles triangle has at least two sides with equal length. The angles opposite those equal sides are of equal measure. Definition 20: "...an isosceles triangle is that which has two of its sides alone equal..." xº xº 25

26 Classifying Triangles 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 26

27 Classifying Triangles Obtuse Triangles An obtuse triangle has one angle which is greater than 90 ºand two acute angles. Definition 21: "...an obtuse-angled triangle is that which has an obtuse angle..." Notice that one angle is greater than 90º, which means that the other two sum to less than 90º; and they are acute.. 27

28 Classifying Triangles Equiangular / Equilateral Triangles An equiangular, or equilateral, triangle has angles of equal measure and sides of equal length. xº 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. xº xº This is a special acute isosceles triangle. 28

29 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. 29

30 10 An isosceles triangle is an equilateral triangle. A B C Sometimes Always Never 30

31 11 An obtuse triangle is an isosceles triangle. A B C Sometimes Always Never 31

32 12 A triangle can have more than one obtuse angle. True False 32

33 13 A triangle can have more than one right angle. True False 33

34 14 Each angle in an equiangular triangle measures 60 True False 34

35 15 An equilateral triangle is also an isosceles triangle True False 35

36 16 This triangle is classified as. (Choose all that apply.) A acute B right C isosceles º 8.6 D obtuse E equilateral 60º º F equiangular G scalene 36

37 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 º 44º

38 18 This triangle is classified as. (Choose all that apply.) A acute B right C isosceles D obtuse E equilateral F equiangular G scalene 38

39 19 This triangle is classified as. Choose all that apply. A acute B right C isosceles D obtuse E equilateral F equiangular G scalene

40 Example Measure and Classify the triangle by sides and angles Math Practice isosceles, Click for acute

41 Example Measure and Classify the triangle by sides and angles Math Practice scalene, Click for obtuse

42 Example Measure and Classify the triangle by sides and angles scalene, Click for acute Math Practice

43 20 Classify the triangle with the given information: Side lengths: 3 cm, 4 cm, 5 cm A Equilateral D Acute B Isosceles E Equiangular C Scalene F G Right Obtuse 43

44 21 Classify the triangle with the given information: Side lengths: 3 cm, 2 cm, 3 cm A Equilateral D Acute B C Isosceles Scalene E F G Equiangular Right Obtuse 44

45 22 Classify the triangle with the given information: Side lengths: 5 cm, 5 cm, 5 cm A B Equilateral Isosceles D Acute E Equiangular C Scalene F Right G Obtuse 45

46 23 Classify the triangle with the given information: Angle Measures: 25, 120, 35 A Equilateral D Acute B Isosceles E Equiangular C Scalene F G Right Obtuse 46

47 24 Classify the triangle with the given information: Angle Measures: 30, 60, 90 A Equilateral D Acute B C Isosceles Scalene E F G Equiangular Right Obtuse 47

48 25 Classify the triangle with the given information: Side lengths: 3 cm, 4 cm, 5 cm Angle measures: 37, 53, 90 A B C Equilateral Isosceles Scalene D E F G Acute Equiangular Right Obtuse 48

49 26 Classify the triangle by sides and angles A B Equilateral Isosceles A 120 B C D E Scalene Acute Equiangular F G Right Obtuse C 49

50 27 Classify the triangle by sides and angles A Equilateral B Isosceles C Scalene D Acute E Equiangular F Right G Obtuse L N M 50

51 28 Classify the triangle by sides and angles A B C Equilateral Isosceles Scalene J 85 D E F G Acute Equiangular Right Obtuse H K 51

52 Triangle Sum Theorem Return to Table of Contents 52

53 Triangle Sum Theorem We can use what we learned about parallel lines to determine the sum of the measures of the angles of any triangle. A Math Practice B C First, let's draw two parallel lines. The first along the base of the triangle and the other through the opposite vertex. 53

54 Triangle Sum Theorem x A y B x y C And extend AB to make it a transversal. Then, let's label some of the angles. 54

55 29 What is the name for the pair of angles labeled x and what is the relationship between them? A outside exterior, they are unequal B alternate interior, they are unequal C alternate interior, they are equal D outside exterior, they are equal Is the same true for the pair of angles labeled y? 55

56 Triangle Sum Theorem Therefore, both angles labeled x are equal and can be called x, and x has the same measure as B. A x B x C Repeat the same process with side AC and find an angle along the upper parallel line equal to angle C 56

57 Triangle Sum Theorem x A y B x y C Let's just re-label the upper angles with A, B and C. 57

58 Triangle Sum Theorem The sum of those angles along that upper parallel line equals 180º, so A + B + C = 180º B A C 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 58

59 Triangle Sum Theorem The measures of the interior angles of a triangle sum to 180 A Math Practice B C Click here to go to the lab titled, "Triangle Sum Theorem" 59

60 Example: Triangle Sum Theorem J Find the measure of the missing angle. 32º K 20º L 60

61 30 What is m B? C 52 A 53 B 61

62 31 What is the measurement of the missing angle? M N 57 L 62

63 32 In ΔABC, if m B is 84 and m C is 36, what is m A? 63

64 33 In ΔDEF, if m D is 63 and m E is 12, find m F. 64

65 Solve for x Example Q (12x+8) P 55 (8x-3) R 65

66 34 Solve for x. 8x R Q 2x 5x Then find: m Q = m R = m S = S 66

67 35 What is the measure of B? B (3x-17) A (x+40) (2x-5) C 67

68 Corollary to Triangle Sum Theorem The acute angles of a right triangle are complementary. B A C 68

69 Proof of Triangle Sum Theorem Corollary B Given: Triangle ABC is a right triangle Prove: Its acute angles, Angles B and C, are complementary A C 69

70 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 Statement B A Reason 1 Triangle ABC is a right triangle? Right triangles contain a right 2 angle.? 3? Interior Angles Theorem 4 m A = 90º? 5 90º + m B + m C = 180º? 6 m B + m C = 90º? C 7? Definition of complementary 70

71 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 Statement B A Reason 1 Triangle ABC is a right triangle? Right triangles contain a right 2 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 C 71

72 38 Which reason applies to step 3? B 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 A Reason 1 Triangle ABC is a right triangle? Right triangles contain a right 2 angle.? 3? Interior Angles Theorem 4 m A = 90º? 5 90º + m B + m C = 180º? 6 m B + m C = 90º? C 7? Definition of complementary 72

73 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 Statement B A Reason 1 Triangle ABC is a right triangle? Right triangles contain a right 2 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 C 73

74 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 Statement B A Reason 1 Triangle ABC is a right triangle? Right triangles contain a right 2 angle.? 3? Interior Angles Theorem 4 m A = 90º? 5 90º + m B + m C = 180º? 6 m B + m C = 90º? C 7? Definition of complementary 74

75 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 Statement B A Reason 1 Triangle ABC is a right triangle? Right triangles contain a right 2 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 C 75

76 42 Which reason applies to step 7? A The measure of a straight angle is 180º B 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 A is a right angle Statement Reason 1 Triangle ABC is a right triangle? Right triangles contain a right 2 angle.? 3? Interior Angles Theorem 4 m A = 90º? 5 90º + m B + m C = 180º? 6 m B + m C = 90º? C 7? Definition of complementary 76

77 Proof of Triangle Sum Theorem Corollary Given: Triangle ABC is a right triangle Prove: Its acute angles, Angles B and C, are complementary A C Statement Reason 1 Triangle ABC is a right triangle Given Right triangles contain a right 2 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 6 m B + m C = 90º 7 The acute angles are complementary B Substitution Property of Equality Subtraction Property of Equality Definition of complementary 77

78 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. 78

79 43 In a right triangle, the two acute angles sum to 90. True False 79

80 44 What is the measurement of the missing angle? M N 57 L 80

81 45 Solve for x. A What are the measures of the three angles? B C 81

82 46 Solve for x. What are the measures of the three angles? 82

83 47 m 1 + m 2 =

84 48 m 1 + m 3 =

85 49 Find the value of x in the diagram 20 x 85

86 Exterior Angle Theorem Return to Table of Contents 86

87 Exterior Angles 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. A xº B C 87

88 Exterior Angles 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. A xº B C 88

89 Exterior Angles The exterior angles at each vertex are congruent, since they are vertical angles. xº A xº B C 89

90 Remote Interior Angles 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. In this case, A and C are the remote interior angles and ABC is the adjacent interior angle. xº B A C 90

91 50 Which are the remote interior angles in this instance? A A & B B A & C C B & C xº A xº B C 91

92 51 If line AB is a straight line, what is the sum of 2 and 1? A 2 1 B 92

93 52 In this diagram, what is the sum of angles P, Q and R? P R Q 93

94 Exterior Angles Theorem 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 or x = m A + m C A D xº B C 94

95 Proof of Exterior Angles Theorem A D xº B C Given: DBA is an exterior angle of ΔABC and A and C are remote interior angles. Prove: m DBA = m A + m C 95

96 1 53 Which reason applies to step 2? A Angles that form a linear pair are supplementary B Definition of complementary A C Interior Angles Theorem D Substitution Property of Equality xº E Definition of a right angle Statement DBA is an exterior angle of ΔABC and A and C are remote interior angles D B Reason Given 2 DBA and ABC are supplementary? 3? Definition of supplementary 4 m A+ m ABC + m C = 180? C 5 m DBA + m ABC = m A + m ABC + m C? 6? Subtraction Property of Equality 96

97 1 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 x E B m DBA + m A = 180 D Statement DBA is an exterior angle of ΔABC and A and C are remote interior angles A Reason 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 C 97

98 1 55 Which reason applies to step 4? A Angles that form a linear pair are supplementary B Definition of complementary A C Interior Angles Theorem D Substitution Property of Equality x E Definition of a right angle Statement DBA is an exterior angle of ΔABC and A and C are remote interior angles D B Reason 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 C 98

99 1 56 Which reason applies to step 5? A Angles that form a linear pair are supplementary B Definition of complementary A C Interior Angles Theorem D Substitution Property of Equality x E Definition of a right angle Statement DBA is an exterior angle of ΔABC and A and C are remote interior angles D B Reason 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 C 99

100 1 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 x E B m DBA + m A = 180 D Statement DBA is an exterior angle of ΔABC and A and C are remote interior angles A Reason 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 C 100

101 1 Proof of Exterior Angles Theorem Given: DBA is an exterior angle of ΔABC and A and C are remote interior angles. Prove: m DBA = m A + m C Statement DBA is an exterior angle of ΔABC and A and C are remote interior angles 2 DBA and ABC are supplementary 3 DBA + m ABC = 180 Reason Given Angles that form a linear pair are supplementary 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 6 m DBA = m A + m C D x B A Substitution Property of Equality Subtraction Property of Equality C 101

102 58 In this case, what must be the relationship between the interior angles of ΔPQR and 1? A m Q = m 1 B C D m 1 = m P m 1 = m Q + m R m 1 = m P + m R P E m 1 = m Q + m P R 1 Q 102

103 59 In this case, what must be the relationship between the interior angles of ΔPQR and 2? A B C D E m Q = m 2 m 2 = m P m 2 = m Q + m R m 2 = m P + m R m 2 = m Q + m P R P 2 Q 103

104 Example: Using the Exterior Angle Theorem R xº P xº 140º Q What is the value of x? 104

105 Example Solve for x and y. 21 x y

106 Example Solve for x and y. 75º xº yº 50º 106

107 60 Solve for x. 60º 55º xº yº 107

108 61 Solve for y. 60º 55º xº yº 108

109 62 Find the value of x. 94º yº 60º 2xº 109

110 63 Find the value of x. 100º yº (2x+3)º 51º 110

111 64 Find the value of x. (3x-5) y (x+2)

112 65 Segment PS bisects RST, what is the value of w? S 25 R wº P T 112

113 Example Find the missing angles in the diagram Teacher Notes

114 66 Find the measure of º º 114

115 67 Find the measure of º º 115

116 68 Find the measure of angle º º 116

117 69 Find the measure of 4. angle º º 117

118 70 Find the the measure of of angle º º 118

119 Inequalities in Triangles Return to Table of Contents 119

120 Inequalities in one Triangle To investigate inequalities in one triangle download the sketch, "inequalities in one triangle" and the worksheet, "inequalities in one triangle" Go to the sketch, "Inequalities in one triangle." Go to the worksheet, "Inequalities in one triangle." Math Practice 120

121 Angle Inequalities in a Triangle The longest side is always opposite the largest angle. The shortest side is always opposite the smallest angle. 121

122 71 Name the longest side of this triangle. A AB B BC C CA D They are all equal A B 85 C 122

123 72 Name the shortest side of this triangle. A AB B BC C CA D They are all equal A B 85 C 123

124 73 Name the shortest side of this triangle. A AB B BC A C CA D They are all equal B 40 C 124

125 74 Name the largest angle of this triangle. A A B B C C D They are all equal A 10 B 14 8 C 125

126 75 Name the smallest angle of this triangle. A A B B C C D They are all equal A 10 B 14 8 C 126

127 76 Name the smallest angle of this triangle. A A B B A C C D They are all equal B 10 C 127

128 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. 128

129 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. Math Practice Move the sides below and try to form a triangle. 129

130 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. Math Practice 130

131 77 What is the maximum length of the third side to form a triangle if the other sides are 4 and 6? 131

132 78 What is the maximum length of the third side to form a triangle if the other sides are 8 and 7? 132

133 79 What is the minimum length of the third side to form a triangle if the other sides are 4 and 6? 133

134 80 What is the minimum length of the third side to form a triangle if the other sides are 7 and 8? 134

135 Similar Triangles Return to Table of Contents 135

136 Congruence Recall that: 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 a b which is read as "a is congruent to b." 136

137 Congruent Line Segments We learned earlier that: Only line segments with the same length are congruent. Also, all congruent segments have the same length. a b c d a b c d 137

138 A B Congruent Angles Recall: Two angles are congruent if they have the same measure. Two angles are not congruent if they have different measures. If m A = m B If m C m D C D A B C D 138

139 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. 139

140 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. 140

141 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. 141

142 Similar Triangles Have Proportional Sides Theorem 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. B E A C D F 142

143 Similar Triangles This is the symbol for similarity So, the symbolic statement for Triangle ABC is similar to Triangle DEF is: ΔABC ~ ΔDEF 143

144 Naming Similar Triangles ΔABC ~ ΔDEF 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 Math Practice And, thereby which are the corresponding, proportional, sides. AB corresponds to DE BC corresponds to EF CA corresponds to FD 144

145 Naming Similar Triangles ΔABC ~ ΔDEF 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 one another. Math Practice 145

146 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. Math Practice 146

147 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. 147

148 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: 1 Statement A and B in ΔABC are to D and E in ΔDEF 2 m A = m D; m B = m E m A+ m B + m C = 180º m D+ m E + m F = 180º m C =180º - (m A + m B) m F =180º- (m D + m E) m C =180º - (m A + m B) m F =180º- (m A + m B) Reason Given Definition of Congruent Angles Triangle Sum Theorem Subtraction Property of Equality Substitution Property of Equality 6 m C = m F Substitution Property of Equality 7 ΔABC and ΔDEF are similar Definition of Similarity 148

149 Side-Side-Side Similarity Theorem 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. 149

150 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 A G H B C 150

151 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. D F A G E H B C 151

152 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. D F A G E H B C 152

153 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 A D G E H B C 153

154 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. A F D G B E H C 154

155 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. 155

156 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. 156

157 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. 157

158 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. B E A C D Click here to see the third sides. F 158

159 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. 159

160 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 E They are not similar x x 160

161 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 161

162 83 Which theorem allows you to prove these two triangles are similar? A Angle-Angle B Side-Angle-Side C Side-Side-Side E They are not similar D They may not be similar

163 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 E They are not similar

164 85 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 4 8 x 3 x 6 164

165 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 E They are not similar x 4 8 x

166 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 166

167 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 167

168 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 168

169 90 Which theorem allows you to prove these two triangles are similar? A Angle-Angle B Side-Angle-Side A C Side-Side-Side D They may not be similar E They are not similar B C Note that BC is parallel to DE. D E 169

170 Similar Triangles Have Proportional 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. B Sides Theorem E A C D F 170

171 Similar Triangles Have Proportional Sides Theorem If two triangles are similar, all of their corresponding sides are in proportion. *While Euclid does prove this theorem, his proof relies on other 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

172 Similar Triangles and Proportionality 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. B E A C D F 172

173 Similar Triangles and Proportionality We also then know that the corresponding sides are proportional. The symbol for proportional is the Greek letter, alpha: α AB α DE, since AB corresponds to DE BC α EF, since BC corresponds to EF AC α DF, since AC corresponds to DF B E A C D F 173

174 Corresponding Sides 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 B E A C D F 174

175 Corresponding Sides 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. B E A C D F 175

176 Corresponding Sides Either approach works; use the one you find easiest. Identify corresponding sides as the sides connecting equal angles or the sides opposite equal angles...you'll get the same result. B E A C D F 176

177 Similar Triangles and Proportionality Another way of saying two sides are proportional is to say that one 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 other triangle. In this case, if ΔABC is k times as big as ΔDEF, then: AB = kde BC = kef AC = kdf B E A C D F 177

178 Similar Triangles and Proportionality 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. B E A C D F 178

179 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 B E A C D F 179

180 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 H L M J N 180

181 Example - Proportional Sides Given that ΔABC is similar to ΔDEF, and given the indicated lengths, find the lengths AB and BC. B E 5 7 A 8 C D 4 F 181

182 Example - Proportional Sides B E 5 7 Math Practice A 8 C Since the triangles are similar we know that the following relationship holds between all the corresponding sides. AB BC AC ED = EF = DF = k 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? D 4 F 182

183 Example - Proportional Sides B E A AC 8 = = k = 2 DF 4 8 C D 5 7 That means that the other two sides of ΔABC will also be twice as large as the corresponding sides of ΔDEF 4 F Math Practice AB BC AC ED = EF = DF = k = 2 How can we write the proportions required to calculate AB and BC? 183

184 Example - Proportional Sides B E 5 7 A C 8 D 4 F AB ED = 2 AB 5 = 2 AB = 10 BC EF = 2 BC 7 = 2 BC =

185 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 =? B E A C D F 185

186 94 Given that ΔJIH is similar to ΔLMN; find the length of LM. 12 I 10 H L 14 M J 5 N 186

187 95 Given that ΔJIH is similar to ΔLMN; find the length of LN. 12 I 10 H L 14 M J 5 N 187

188 96 Given that BC is parallel to DE and the given lengths, find the length of DE. A 8 4 B 6 C D E 188

189 97 Given that BC is parallel to DE and the given lengths, find the length of DB. A 7 B 3 C D 9 E 189

190 Example - Similarity & Proportional Sides Determine if the triangles are similar. If they are similar write a similarity statement. If they are not similar, explain why. P R 6 B D 9 K L 190

191 Example - Similarity & Proportional Sides P R 6 B 12 D 18 9 K L Math Practice 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. 191

192 Example - Similarity & Proportional Sides P 6 R B D 9 Side of ΔPDK Length Side of ΔBRL Length Ratio K DK 9 BR PD 15 RL PK 18 BL 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. L 192

193 98 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 6 B D 9 K L 193

194 99 If these triangles are similar, enter the constant of proportionality, k, between the larger and smaller triangle. If they are not, enter zero. R 1 S T X 4 52 Y 6 2 Z 194

195 100 If these triangles are similar, enter the constant of proportionality, k, between the larger and smaller triangle. If they are not, enter zero. R C P S B D 195

196 Side Splitter Theorem Any line parallel to a side of a triangle will form a triangle which is similar to the first triangle. A 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. D B C E 196

197 Proof of Side Splitter Theorem Given: BC is parallel to DE A Prove: ΔABC ~ ΔADE. B C D E 197

198 101 What is the reason for step 2? A 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. Statement D B Reason 1 BC is parallel to DE Given 2 ABC D; ACB E? 3 A A? 4 ΔABC ~ ΔADE? C E 198

199 102 What is the reason for step 3? A 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. Statement D B Reason 1 BC is parallel to DE Given 2 ABC D; ACB E? 3 A A? 4 ΔABC ~ ΔADE? C E 199

200 103 What is the reason for step 4? A A Angle-Angle Similarity Theorem B Side-Side-Side Similarity Theorem C Reflexive Property of Congruence D E 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. Statement B Reason 1 BC is parallel to DE Given 2 ABC D; ACB E? 3 A A? 4 ΔABC ~ ΔADE? C 200

201 Proof of Side Splitter Theorem A Given: BC is parallel to DE Prove: ΔABC ~ ΔADE B C Statement D 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 E 201

202 Proof of Side Splitter Theorem A Now, we know that ΔABC ~ ΔADE. The remaining steps of this proof use the properties of proportional triangles and equality. B C Given: ΔABC ~ ΔADE D E Prove: BD AB = CE AC 202

203 Statement Reason 1 ΔABC ~ ΔADE Given 2 AD = AE AB AC 3 AB + BD = AD, AC + CE = AE? Proof of Side Splitter Theorem The table below will be used to answer the next 5 Response Questions AB + BD = AC + CE AB AC AB + BD = AC + CE AB AB AC AC 1 + BD = 1 + AB BD = CE AB AC CE AC?? Addition Property of Fractions ( e.g = = 5 ) ?? 203

204 104 What is the reason for step 2? A 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 D B C E 204

205 105 What is the reason for step 3? A 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 D B C E 205

206 106 What is the reason for step 4? A 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 D B C E 206

207 107 What is the reason for step 6? A 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 D B C E 207

208 108 What is the reason for step 7? A 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 D B C E 208

209 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 Proof of Side Splitter Theorem Given: ΔABC ~ ΔADE AB + BD = AC + CE AB AC AB + BD = AC + CE AB AB AC AC 1 + BD = 1 + AB BD = CE AB AC CE AC Prove: BD AB CE AC Substitution Property of Equality Addition Property of Fractions ( e.g = = 5 ) Simplify the Equation Subtraction Property of Equality = 209

210 Converse of Side Splitter Theorem A 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. B C D E 210

211 109 Find the value of x to prove that AB is parallel to ER. B 27 E 18 D 12 R x A 211

212 110 Find the value of x to prove that FC is parallel to MN. F 6 M C x N 9 8 J 212

213 111 Find the value of y y 6 213

214 112 Find the value of y y 4 214

215 113 Find the value of y y

216 PARCC Sample Question and Applications Return to Table of Contents 216

217 114 The figure ΔABC ~ ΔDEF with side lengths as indicated. What is the value of x? F 5 9 D C E 7 27 x A 21 B From PARCC EOY sample test 217

218 Using Similar Triangles 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? 218

219 Using Similar Triangles How can we find the distance across the Grand Canyon? Math Practice Grand Canyon National Park, AZ 219

220 Using Similar Triangles 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. 220

221 Using Similar Triangles 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. 221

222 Using Similar Triangles How can you prove that ΔABC ~ ΔEDC? How can you find the distance across the Grand Canyon? 222

223 Using Similar Triangles DCE BCA CDE CBA ΔABC ~ ΔEDC Why? Why? Why? 223

224 Using Similar Triangles How do you find d? Write a statement of proportionality that uses d. click 224

225 Using Similar Triangles How can we find the height of the Washington Monument when there are no shadows? 225

226 Using Similar Triangles 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. Math Practice 226

227 Using Similar Triangles In Physics, angle of reflection = angle of incidence angle of reflection angle of incidence reflected ray incident ray surface 227

228 Using Similar Triangles 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? 228

229 Using Similar Triangles CAB EAD ACB EAD ΔABC ~ ΔADE Why? Why? Why? 229

230 Using Similar Triangles How do you find h? Write a statement of proportionality that uses h. click 230

231 115 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 g C ft D 18.9 ft 15 ft 5 ft You 5 ft 3 in 231

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 w = w = 63 9 w = = w