1. Chapter 4 Congruent Triangles

2. 4.1 Triangles and Angles

3. Parts of Triangles Vertex Vertex Points joining the sides of a triangle Points joining the sides of a triangle Adjacent Sides Adjacent Sides Sides that share a common vertex Sides that share a common vertex

4. Classification by Sides Equilateral Equilateral 3 congruent sides 3 congruent sides Isosceles Isosceles At least 2 congruent sides At least 2 congruent sides Scalene Scalene No congruent sides No congruent sides

5. Classification by Angles Acute Acute 3 acute angles 3 acute angles Equiangular Equiangular 3 congruent angles 3 congruent angles Right Right 1 right angle 1 right angle Obtuse Obtuse 1 obtuse angle 1 obtuse angle

6. Parts of Isosceles Triangles Legs Legs The sides that are congruent. The sides that are congruent. Base Base The non-congruent side. The non-congruent side.

7. Isosceles triangles Base angles are congruent. Base angles are congruent. Base Base angles legs Vertex angle

8. Parts of Right Triangles Hypotenuse Hypotenuse The side that is opposite the right angle. It is always the longest side. The side that is opposite the right angle. It is always the longest side. Legs Legs The sides that form the right angle The sides that form the right angle

9. Right Triangles leg hypotenuse

10. Interior Angles The angles on the inside of a triangle. The angles on the inside of a triangle.

11. Triangle Sum Conjecture The sum of the measures of the angles in every triangle is 180 . The sum of the measures of the angles in every triangle is 180 .

12. Example Find the measure of each angle. 2x + 10 xx + 2

13. Exterior Angles The angles that are adjacent to the interior angles The angles that are adjacent to the interior angles The exterior angles always add to equal 360° The exterior angles always add to equal 360°

14. Definitions Exterior AngleAdjacent Exterior AngleAdjacent Interior Interior Angle Angle Remote Interior Angles

15. Exterior Angles of a Triangle Use your straightedge to draw a triangle. Use your straightedge to draw a triangle. Extend one side out as shown Extend one side out as shown A B C x a b c

16. Exterior Angles of a Triangle Trace angles a and b onto a transparency so that they are adjacent. Trace angles a and b onto a transparency so that they are adjacent. How does this compare to angle x? How does this compare to angle x? A B C x a b c a b

17. Triangle Exterior Angle Conjecture The measure of the exterior angle of a triangle is equal to the sum of the measures of the remote interior angles The measure of the exterior angle of a triangle is equal to the sum of the measures of the remote interior angles A B C x = a + b a b c

18. Example Find the missing measures 80° 53°

19. Example Find the missing measures 60° 120°

20. Example Page 199 #37 Page 199 #37 (2x – 8)° x°x°31°

21. 4.2 Congruence and Triangles

22. Terms Congruent Congruent Figures that are exactly the same size and shape are congruent Figures that are exactly the same size and shape are congruent Corresponding angles Corresponding angles The angles that are in corresponding positions are congruent The angles that are in corresponding positions are congruent Corresponding sides Corresponding sides The sides that are in corresponding positions are congruent The sides that are in corresponding positions are congruent

23. Naming Congruent Figures When a congruence statement is made it is important to match up corresponding parts. When a congruence statement is made it is important to match up corresponding parts.

24. Third Angle Theorem If two angles in one triangle are equal to two angles in another triangle, then the third angles in each triangle are also equal. If two angles in one triangle are equal to two angles in another triangle, then the third angles in each triangle are also equal.

25. Examples 1 (page 205) ΔLMN ΔPQR ΔLMN  ΔPQR 105° ML N 45° P R Q What is the measure of:  P  M  R  N Which side is congruent to segment QR Segment LN

26. Example 2 Given  ABC   PQR, find the values of x and y. A B C P Q R 85° 50° (6y – 4)° (10x + 5)°

27. 4.3 Proving Triangles Congruent SSS and SAS

28. Warm-Up Complete the following statement  BIG  B I G R A T

29. Definitions included angle included angle An angle that is between two given sides. An angle that is between two given sides. included side included side A side that is between two given angles. A side that is between two given angles.

30. Example 1 Use the diagram. Name the included angle between the pair of given sides. Use the diagram. Name the included angle between the pair of given sides. K P JL

31. Triangle Congruence Shortcut SSS SSS If the three sides of one triangle are congruent to the three sides of another triangle, then the triangles are congruent. If the three sides of one triangle are congruent to the three sides of another triangle, then the triangles are congruent.

32. Triangle Congruence Shortcuts SAS SAS If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent. If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent.

33. Example 2 Complete the congruence statement. Complete the congruence statement. Name the congruence shortcut used. Name the congruence shortcut used. S T U V W  STW  

34. Example 3 Determine if the following are congruent. Determine if the following are congruent. Name the congruence shortcut used. Name the congruence shortcut used. H I J L M N  HIJ   LMN

35. Example 4 Complete the congruence statement. Complete the congruence statement. Name the congruence shortcut used. Name the congruence shortcut used. B O X C A R  XBO  

36. Example 5 Complete the congruence statement. Complete the congruence statement. Name the congruence shortcut used. Name the congruence shortcut used.  SPQ   S P Q T

37. 4.4 Proving Triangles Congruent ASA and AAS

38. Triangle Congruence Shortcuts ASA ASA If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent. If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent.

39. Triangle Congruence Shortcuts SAA SAA If two angles and a non-included side of one triangle are congruent to the corresponding angles and side of another triangle, then the two triangles are congruent. If two angles and a non-included side of one triangle are congruent to the corresponding angles and side of another triangle, then the two triangles are congruent.

40. Example 1 Complete the congruence statement. Complete the congruence statement. Name the congruence shortcut used. Name the congruence shortcut used. Q U AD  QUA  

41. Example 2 Complete the congruence statement. Complete the congruence statement. Name the congruence shortcut used. Name the congruence shortcut used. M R N Q P  RMQ  

43. Example 3 Determine if the following are congruent. Determine if the following are congruent. Name the congruence shortcut used. Name the congruence shortcut used. A B C F E D  ABC   FED

44. 4.6 Isosceles, Equilateral, and Right Triangles

45. Warm-Up 1 Find the measure of each angle. 90° 30° 60° a b

46. Warm-Up 2 Find the measure of each angle.  110   150   90 

47. The base angles of an isosceles triangle are congruent. The base angles of an isosceles triangle are congruent. If a triangle has at least two congruent angles, then it is an isosceles triangle. If a triangle has at least two congruent angles, then it is an isosceles triangle. If the sides are congruent then the base angles are congruent. If the sides are congruent then the base angles are congruent. Isosceles triangles

48. Example 1 35° x

49. Example 2 15° b a

50. Example 3 Find each missing measure 63° 10 cm m n p

51. Equilateral Triangles If a triangle is equilateral, then it is equiangular. If a triangle is equilateral, then it is equiangular. If a triangle is equiangular, then it is equilateral. If a triangle is equiangular, then it is equilateral.

52. Hypotenuse-Leg (HL) If the hypotenuse and the leg of a right triangle are congruent to the hypotenuse and leg of a second right triangle, then the two triangles are congruent. If the hypotenuse and the leg of a right triangle are congruent to the hypotenuse and leg of a second right triangle, then the two triangles are congruent.

53. Example 4 Find the value of x Find the value of x 12 in 2x in

54. Example 5 Find the value of x and y. Find the value of x and y. y x

55. Example 6 Find the value of x and y. Find the value of x and y. 75° x°x° y°y° x°x°

56. Chapter 8 Similarity

57. 8.1 Ratio and Proportion

58. Ratios The ratio of a to b can be written as The ratio of a to b can be written as a/b a/b a : b a : b The denominator cannot be zero The denominator cannot be zero

59. Simplifying Ratios Ratios should be expressed in simplified form Ratios should be expressed in simplified form 6:8 = 3:4 6:8 = 3:4 Before reducing, make sure that the units are the same. Before reducing, make sure that the units are the same. 12in : 3 ft 12in : 3 ft 12in : 36 in 1: 3

60. Examples (page 461) Simplify each ratio Simplify each ratio 10.16 students 24 students 12.22 feet 52 feet 18.60 cm 1 m

61. Examples (page 461) Simplify each ratio Simplify each ratio 20. 2 mi 3000 ft 24. 20 oz. 24. 20 oz. 4 lb There are 5280 ft in 1 mi. There are 16 oz in 1 lb.

62. Examples (page 461) Find the width to length ratio Find the width to length ratio14.16. 20 mm 16 mm 2 ft 12 in.

63. Using Ratios Example 1 The perimeter of the isosceles triangle shown is 56 in. The ratio of LM : MN is 5:4. Find the length of the sides and the base of the triangle. The perimeter of the isosceles triangle shown is 56 in. The ratio of LM : MN is 5:4. Find the length of the sides and the base of the triangle. N L M

64. Using Ratios Example 2 The measures of the angles in a triangle are in the extended ratio 3:4:8. Find the measures of the angles The measures of the angles in a triangle are in the extended ratio 3:4:8. Find the measures of the angles 3x 4x 8x

65. Using Ratios Example 3 The ratios of the side lengths of ΔQRS to the corresponding side lengths of ΔVTU are 3:2. Find the unknown lengths. The ratios of the side lengths of ΔQRS to the corresponding side lengths of ΔVTU are 3:2. Find the unknown lengths. S T R Q V U 2 cm 18 cm

66. Proportions Proportion Proportion Ratio = Ratio Ratio = Ratio Fraction = Fraction Fraction = Fraction Solving Proportions Solving Proportions Cross multiply Cross multiply Let the means equal the extremes Let the means equal the extremes

67. Properties of Proportions Cross Product Property Cross Product Property Reciprocal Property Reciprocal Property

68. Solving Proportions Example 1

69. Solving Proportions Example 2

70. Solving Proportions Example 3 A photo of a building has the measurements shown. The actual building is 26 ¼ ft wide. How tall is it? 1 7/8 in 2.75 in

71. 8.2 Problem solving in Geometry with Proportions

72. Properties of Proportions

73. Example 1 Tell whether the statement is true or false Tell whether the statement is true or false A. A. B. B.

74. Example 2 In the diagram In the diagram Find the length of LQ. P M 6 N Q 15 13 L5

75. Geometric Mean Geometric Mean Geometric Mean The geometric mean between two numbers a and b is the positive number x such that The geometric mean between two numbers a and b is the positive number x such that

76. Example 3 Find the geometric mean between 35 and 175. Find the geometric mean between 35 and 175.

77. Example 4 You are building a scale model of your uncle’s fishing boat. The boat is 62 ft long and 23 ft wide. The model will be 14 in. long. How wide should it be? You are building a scale model of your uncle’s fishing boat. The boat is 62 ft long and 23 ft wide. The model will be 14 in. long. How wide should it be?

78. 8.3 Similar Polygons

79. Polygons are similar if and only if the corresponding angles are congruent the corresponding angles are congruentand the corresponding sides are proportional. the corresponding sides are proportional.

80. Similar figures are dilations of each other. (They are reduced or enlarged by a scale factor.) Similar figures are dilations of each other. (They are reduced or enlarged by a scale factor.) The symbol for similar is  The symbol for similar is 

81. Example 1 Determine if the sides of the polygon are proportional. 12 m 8 m 6 m

82. Example 2 Determine if the sides of the polygon are proportional. 15 m 5 m 9 m 12 m 3 m 4 m

83. Example 3 Find the missing measurements. HAPIE  NWYRS H A P IE N W Y RS 6 54 18 24 21 AP = EI = SN = YR =

84. Example 4 D Find the missing measurements. QUAD  SIML A U Q M I S L 20 25 125º 8 95º 65º QD = MI = m  D = m  U = m  A = 12

85. 8.4/8.5 Similar Triangles

86. To be similar, corresponding sides must be proportional and corresponding angles are congruent. To be similar, corresponding sides must be proportional and corresponding angles are congruent.

87. Similarity Shortcuts AA Similarity Shortcut If two angles in one triangle are congruent to two angles in another triangle, then the triangles are similar.

88. Similarity Shortcuts SSS Similarity Shortcut If three sides in one triangle are proportional to the three sides in another triangle, then the triangles are similar.

89. Similarity Shortcuts SAS Similarity Shortcut If two sides of one triangle are proportional to two sides of another triangle and their included angles are congruent, then the triangles are similar.

90. Similarity Shortcuts We have three shortcuts: AASASSSS

91. Example 1 4 g 7 6 9 10.5

92. Example 2 h 32 24 50 k 30

93. Example 3 42 m 36 24

94. 1. A flagpole 4 meters tall casts a 6 meter shadow. At the same time of day, a nearby building casts a 24 meter shadow. How tall is the building? 4m4m 6m 24m

95. 2. Five foot tall Melody casts an 84 inch shadow. How tall is her friend if, at the same time of day, his shadow is 1 foot shorter than hers? 2. Five foot tall Melody casts an 84 inch shadow. How tall is her friend if, at the same time of day, his shadow is 1 foot shorter than hers?

96. 3. A 10 meter rope from the top of a flagpole reaches to the end of the flagpole’s 6 meter shadow. How tall is the nearby football goalpost if, at the same moment, it has a shadow of 4 meters? 10m 6m 4m

97. 4. Private eye Samantha Diamond places a mirror on the ground between herself and an apartment building and stands so that when she looks into the mirror, she sees into a window. The mirror is 1.22 meters from her feet and 7.32 meters from the base of the building. Sam’s eye is 1.82 meters above the ground. How high is the window? 1.227.32 1.82

98. 8.6 Proportions and Similar Triangles

99. Proportions Using similar triangles missing sides can be found by setting up proportions. Using similar triangles missing sides can be found by setting up proportions.

100. Theorem Triangle Proportionality Theorem Triangle Proportionality Theorem If a line parallel to one side of a triangle intersects the other two sides, then it divides the two sides proportionally. If a line parallel to one side of a triangle intersects the other two sides, then it divides the two sides proportionally. Q S T R U

101. Theorem Converse of the Triangle Proportionality Theorem Converse of the Triangle Proportionality Theorem If a line divides two sides of a triangle proportionally, then it is parallel to the third side. If a line divides two sides of a triangle proportionally, then it is parallel to the third side. Q S T R U

102. Example 1 In the diagram, segment UY is parallel to segment VX, UV = 3, UW = 18 and XW = 16. What is the length of segment YX? In the diagram, segment UY is parallel to segment VX, UV = 3, UW = 18 and XW = 16. What is the length of segment YX? U Y V W X

103. Example 2 Given the diagram, determine whether segment PQ is parallel to segment TR. Given the diagram, determine whether segment PQ is parallel to segment TR. P 9 T S 26 Q 9.75 24 R

104. Theorem If three parallel lines intersect two transversals, then they divide the transversals proportionally. If three parallel lines intersect two transversals, then they divide the transversals proportionally.

105. Theorem If a ray bisects an angle of a triangle, then it divides the opposite side into segments whose lengths are proportional to the lengths of the other two sides. If a ray bisects an angle of a triangle, then it divides the opposite side into segments whose lengths are proportional to the lengths of the other two sides.

106. Example 3 In the diagram,  1  2  3, AB =6, BC=9, EF=8. What is x? In the diagram,  1   2   3, AB =6, BC=9, EF=8. What is x? A E D x 9 B 6 F 8 3 2 C 1

107. Example 4 In the diagram,  LKM  MKN. Use the given side lengths to find the length of segment MN. In the diagram,  LKM   MKN. Use the given side lengths to find the length of segment MN. 3 K M L 15 17 N

108. 5. Juanita, who is 1.82 meters tall, wants to find the height of a tree in her backyard. From the tree’s base, she walks 12.20 meters along the tree’s shadow to a position where the end of her shadow exactly overlaps the end of the tree’s shadow. She is now 6.10 meters from the end of the shadows. How tall is the tree? 1.82 12.20 6.10