Unit 4 Congruent Triangles.notebook. Geometry. Congruent Triangles. AAS Congruence. Review of Triangle Congruence Proofs.

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1 Geometry Congruent Triangles AAS Congruence Review of Triangle Congruence Proofs Return to Table 1

2 Side opposite Side Side the sides of triangles Adjacent Sides - two sides sharing a common vertex leg leg leg leg In a right triangle, the hypotenuse is the side If an isosceles triangle has 3 congruent sides, it - the line segments that make up a polygon Vertex (vertices) Acute Triangle Obtuse triangle - 3 congruent angles Isosceles Triangle - At least two congruent sides Scalene triangle - No congruent sides 2

10 0 20 30 40 160 170 180 150 50 140 60 130 120 70 110 80 100 90 90 0 100 80 70 110 60 120 50 130 40 30

3 Unit 4 Congruent Triangles.notebook points. A triangle can be classified by its sides and angles. Isosceles Acute Equiangular (also acute) click click click click click right Measure and Classify the triangles by sides and angles

4 Unit 4 Congruent Triangles.notebook Measure and Classify the triangles by sides and angles Side lengths: 3 cm, 4 cm, 5 cm Equilateral Right Side lengths: 3 cm, 2 cm, 3 cm Equilateral Right 4

5 Side lengths: 5 cm, 5 cm, 5 cm Equilateral Right Equilateral Right Equilateral Right 5

6 Equilateral Right Side lengths: 3 cm, 4 cm, 5 cm Equilateral Right 8 Side lengths: 3 cm, 3 cm, 3 cm Equilateral Right 6

7 9 Classify the triangle by sides and angles Equilateral Right 10 Classify the triangle by sides and angles Equilateral Right 11 Classify the triangle by sides and angles Equilateral Right 7

8 12 An isosceles triangle 13 is an isosceles triangle. 14 A triangle can have more than one obtuse angle. 8

9 15 A triangle can have more than one right angle. 16 Each angle in an equiangular triangle measures An equilateral triangle is also an isosceles triangle 9

10 Return to Table The measures of the interior angles of a triangle sum to 180 its three interior angles is 180 Why is this true? Find the measure of the missing angle Theorem T1. The Triangle Sum Theorem says that the interior 10

11 18 What is the measurement of the missing angle? m B = 19 What is the measurement of the missing angle? m N = 20 What is the measure of the missing angle? 11

12 21 what is the m A? 22 what is the m F? We can solve more "complicated" problems using the Triangle Sum Theorem. Solve for x 12

13 23 Solve for x in the diagram. What is m Q m R m S 24 Since T1. the Triangle Sum Theorem says the interior Recall: two angles that add up to 90 are called 13

14 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. Since this is a right triangle, we can use the Corollary to the Triangle Sum Theorem which says the two acute angles are 25 In a right triangle, the two acute angles sum to What is the measurement of the missing angle? 14

15 27 Solve for x A B C Challenge Click to reveal 28 Solve for x D E Challenge Click to reveal F 29 In the right triangle given, what is the measurement of each H G J 15

17 Exterior Angle Theorems Exterior angles are adjacent to the interior angles. Exterior angles and interior angles together form a straight line. The sum of an exterior angle and an interior angle is 180 degrees. 17

18 18

19 Proof of the Exterior Angle Theorem Q Solve for x using the Exterior Angle Theorem marked x, is equal to the two nonadjacent interior angles. What does x + y have to equal? 180 o click click 19

20 click click click click click click click 33 Solve for the exterior angle, x. 20

21 34 B 35 Find the value of x using the Exterior Angles Theorem? A 34 B 17 C 60 D Find the value of y using the Exterior Angles Theorem? A 34 B 17 C 60 D 86 21

22 37 Using the Exterior Angles Theorem, find the value of x. A 100 B 51 C 46 D What is the value of Y? A 80 B 40 C 51 D Find the value of x. A 40 B 37.5 C 20 D 10 22

23 40 PS bisects RST, what is the value of w? A 100 B 110 C 115 D Find the measure of angle 1. 23

24 42 43 Find the measure of angle Find the measure of angle 4. 24

25 45 Find the measure of angle 5. Isosceles Return to Table at least If an isosceles triangle has - two congruent sides are called the two angles adjacent to the base are the base angles 25

26 If two sides of a triangle are congruent, the angles opposite them are congruent. Corollary to BAT (T3) Find the values of x & y in the isosceles triangle below. x = 44; Base Angles are Congruent y = 180; Triangle Sum Th. y + 88 = 180 y = 92 Find the values of x & y in the isosceles triangle below. x = y; Base Angles are Congruent x + y + 52 = 180; Triangle Sum Th. x + x + 52 = 180; Substitution 2x + 52 = 180 2x = 128 x = Solve for the measurements of the angles x and y 26

27 47 Solve for x and y. 48 What are the measurements of the base angles? 49 The vertex angle of an isosceles triangle is 38. What is a possible measurement for the base angles? 27

28 Corollary to Converse of the BAT (T4) 50 What is the measurement of FD? 51 Classify the triangle by sides and angles equilateral equiangular obtuse right 28

29 52 Classify the triangle by sides and angles equilateral equiangular obtuse right 53 Classify the triangle by sides and angles equilateral equiangular obtuse right 54 Classify the triangle by sides and angles equilateral equiangular obtuse right 29

30 Find the value of x and y 1. First, consider the top triangle. The 3 marks indicate this is an equilateral triangle y 4. Two adjacent angles whose non-shared sides form a straight angle 5. Two adjacent angles whose non-shared sides form a straight line are a linear pair. 7. Using the Base Angles Theorem (T3) and the Triangle Sum theorem (T1), we can 55 30

31 56 57 Solve for x in the diagram. 3x & Triangles Return to Table 31

32 if they have the exact size and shape The two triangles are congruent, write: Answer 32

33 Part Corresponding Sides Corresponding Angles 58 What is the corresponding part to J JKL =~ PQR 33

34 59 What is the corresponding part to Q JKL =~ PQR 60 What is the corresponding part to QP JKL =~ PQR 61 Write a congruence statement for the two triangles BVC =~ XCZ XCB =~ BCX VBC =~ ZXC CBV =~ CZX 34

35 62 Complete the congruence statement XYZ = ~ XWZ ZWX WXZ ZXW W What else can be marked congruent? T5. Third Angles Theorem If two angles of a triangle are congruent to two angles of another triangle, then the third angles are congruent. o degrees, Click to reveal then 40 o & m R = m U = 80 o W Find the value of x. Theorem (T5), we know 2) The m B is easy to find with the Triangle Sum Theorem (T1), 3) Substitute to find x 35

36 63 What is the measurement of J 64 Solve for x 65 Find the value of x. 36

37 Return to Table From the Congruence and Triangles section, you learned that two triangles are congruent if the 3 3 corresponding pairs of angles are However, we do not always need all 6 pieces of information to prove 2 triangles congruent. 37

38 If three sides of one triangle are congruent to three sides of another triangle, then the two triangles are congruent. So, we have: 38

39 66 ~ You need to be very careful that you get the corresponding congruent parts in the correct order CAB is not congruent to HKJ 67 ~ 68 ~ 39

40 Return to Table Side-Angle-Side (SAS) Congruence If two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then the two triangles are congruent. 40

41 69 triangle? JKL, sides KL and JK 70 ~ 41

42 71 ~ 72 ~ ~ ~ ~ ~ 73 What type of congruence exists between the two triangles? Not congruent 42

43 74 What type of congruence exists between the two triangles? Not congruent 75 What type of congruence exists between the two triangles? Not congruent 76 What type of congruence exists between the two triangles? Not congruent 43

44 77 What type of congruence exists between the two triangles? Not congruent 78 What type of congruence exists between the two triangles? Not congruent Return to Table 44

Angle-Side-Angle (ASA) Congruence If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the two triangles are

45 Angle-Side-Angle (ASA) Congruence If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent. R are congruent Second: if it is not already marked, check and mark the diagram Third: check your congruence postulates - what piece of it need to be for your chosen congruence? 45

46 79 What is the included side for X and W? W 80 What is the included side for X and Y W 81 congruence between the two triangles? 46

47 82 congruence between the two triangles? What type of congruence exists between the two triangles? Not congruent 47

48 marking the diagram pulling the triangles apart (when needed) makes it much easier to understand the 85 What type of congruence exists between the two triangles? Not congruent Pull click to the reveal triangles apart! Mark click to reveal the congruent parts! Are there any common sides/angles (look for click to reveal letters that repeat)? 86 What type of congruence exists between the two triangles? Not congruent click to reveal 48

49 87 What type of congruence exists between the two triangles? Not congruent Pull click to the reveal triangles apart! Mark click to reveal the congruent parts! Are there any common sides/angles (look for click to reveal letters that repeat)? 88 What type of congruence exists between the two triangles? Not congruent At the intersection of two line you always have angles. 89 What type of congruence exists between the two triangles? SAS ASA 49

50 90 What type of congruence exists between the two triangles? SAS ASA part. AAS Congruence Return to Table Theorem (T7): Angle-Angle-Side (AAS) Congruence If two angles and the nonincluded side of one triangle are congruent to two angles and the corresponding nonincluded side of another triangle, then the two triangles are congruent. 50

51 Why is AAS a Theorem? The Triangle Sum Theorem (T1) allows us to find the So, by AAS, congruence statement? 91 triangles? Not Congruent 51

52 92 triangles? Not Congruent 93 triangles? Not Congruent 94 triangles? W Not Congruent 52

53 95 triangles? Not Congruent 96 triangles? Not Congruent 97 triangles? Not Congruent 53

54 98 triangles? Not Congruent Return to Table Theorem (T8): If the hypotenuse and a leg of one right triangle are equal to the corresponding hypotenuse and leg of another right triangle, then the two triangles are congruent. 54

55 c HL Congruence theorem applies when the corresponding case, the two right triangles are congruent. Are the two triangles congruent? 99 triangles? Given: QS = ~ XZ RS ~ = YZ Not congruent Click to reveal 55

56 100 triangles? Not congruent If they are congruent what is the congruence statement? 101 triangles? Not congruent If they are congruent what is the congruence statement? 102 triangles? W Not congruent If they are congruent what is the congruence statement? 56

57 103 triangles? W Not congruent If they are congruent what is the congruence statement? 104 triangles? Not congruent If they are congruent what is the congruence statement? 105 triangles? Not congruent If they are congruent what is the congruence statement? 57

58 106 triangles? Not congruent If they are congruent what is the congruence statement? 107 triangles? Not congruent If they are congruent what is the congruence statement? 108 triangles? What angles are congruent when parallel lines are cut by a transversal? Not congruent If they are congruent what is the congruence statement? 58

59 109 triangles? Not congruent If they are congruent what is the congruence statement? 110 triangles? Not congruent If they are congruent what is the congruence statement? Return to Table 59

60 ~ ~ ~ 1) Statements ~ ~ ~ Reasons 1) Given 2) AFK = ~ BGK 2) SSS Postulate Solution (flow proof): HF ~ = HJ Given FG = ~ JK Given H is the midpoint of GK. Given FGH = ~ SSS GH ~ = KH JKH Def. of midpoint justified by the reasons on the right-side column. As we read down the table, we can see the thought process laid out. Statements 60

61 Statements, AC bisects BCD 1. Given click click Statements 61

62 H is the midpoint of GJ click click click C A D B Statements Statements lines 62

63 Given: R is the midpoint of QS, PQR and TSR are right 's, PR = ~ TR click A ~ ~ C E D Statements 1) 2) 3) 4) 5) Reasons 1) 2) 3) 4) 5) Def. of midpoint SSS ~ Def. of midpoint ~ Given ~ Given ~ B E is the midpoint of AB and CD orresponding ongruent ongruent Return to Table 63

64 CPCTC C arts of C riangles are C Sometimes, our goal is not to prove two triangles congruent, but to that some other property is true. says that if two or more triangles are congruent by: then all of their corresponding parts are also congruent. or two angles are corresponding parts 2. Prove that the two triangles are congruent 3. State that the two parts are congruent, using as the reason: orresponding ongruent ongruent" 111 you 64

65 112 you 113 you you 65

66 Statements 3. C is the midpoint of AD Given DB bisects ABC ABD = ~ CBD We are given that click BCA ~ = DCE, BC = ~ CD, and B and D are right angles. Since all right angles are congruent, B ~ = D. With the congruent angles and segments, we can conclude that ABC = ~ EDC by ASA. Therefore, BA = ~ DE by CPCTC. 66

67 W Statements 7. If alt. int. 's =, ~ then lines Triangle Coordinate Proofs Return to Table A coordinate proof places a triangle or, any other geometric figure, into a coordinate plane. - the geometric postulates, theorems, and properties, and The only thing that changes from the proofs we have done Formula to calculate side and segment lengths. 67

68 d = and is the point with coordinates Statements A(4,1), B(5,6), and C(1,3) forms an isosceles right triangle d = 68

69 Continued... d = After we plot the points, we can see that it forms a triangle. Review of Triangle Congruence Proofs Return to Table of Contents 69

70 Objective: Prove triangle congruence using triangle congruence postulates and theorems Given: S Prove: statements reasons 3 K L 4 T 1 2 R 70

4.1 TRIANGLES AND ANGLES

4.1 TRIANGLES AND ANGLES polygon- a closed figure in a plane that is made up of segments, called sides, that intersect only at their endpoints, called vertices Can you name these? triangle- a three-sided