1. Congruent Triangles
Unit 3

2. Bellringer
Find the equation of the line that has a slope of -5 and a y-intercept of 7.
Find the equation of the line that has a slope of 2/3 and goes through the point (-9,-3).

3. Apply Triangle Sum Properties
4.1
I CAN classify triangles and find measures of their angles.

4. Triangle – a polygon with three sides

5. Example 1: Classify the triangular shape of the support beams in the diagram by its sides and by measuring its angles.

6. Ways to Measure Side Lengths and Angles
Ruler - Sides
Protractor – Angles
Distance Formula – Sides
Right Angles - Slopes

7. Example 2:Classifying Triangles
Classify ABC by its sides and by its angles.

8. Vocabulary Interior Angles: Angles inside the polygon
Exterior Angles: The angles that form linear pairs with the interior angles

9. Theorem 4.1 – Triangle Sum Theorem
The sum of the measures of the interior angles of a triangle is 180⁰.

10. Example 4:
Find the measure of each interior angle of ABC, where m∠A = x⁰, m∠B = 2x⁰, and m∠C = 3x⁰.
Example 5:
Find the measures of the acute angles of the right triangle in the diagram shown.

11. Corollary to a theorem – a statement that can be proved easily using the theorem
Corollary to the Triangle Sum Theorem
The acute angles of a right triangle are complementary.
m∠A + m∠B = 90⁰ A C B

12. Theorem 4.2 Exterior Angle Theorem
The measure of an exterior angle of a triangle is equal to the sum of the measures of the two nonadjacent interior angles.

13. Example 6
Find m∠DEF.

14. Example 7:
Find the measure of in the diagram shown.

15. Exit Slip

16. Class Work/Homework
Page 221
#1-6,8-10,14-16

17. Congruence and Triangles
4.2
I CAN define congruent figures that have the same shape and size.

18. Congruent Figures – all parts of one figure are congruent to the corresponding parts of the other figure

19. Congruence Statements
Always list the corresponding vertices in the same order
Can be written in more than one way

20. Example 1
Write a congruence statement for the triangles shown. Identify all pairs of congruent corresponding parts.

21. Example 2 In the diagram, ABCD ≅ FGHK. Find the value of x.
Find the value of y.

22. Example 3
If you divide the wall into orange and blue sections along JK , will the sections of the wall be the same size and shape? Explain.

23. Third Angles Theorem
If two angles of one triangle are congruent to two angles of another triangle, then the third angles are also congruent.

24. Example 4
Find m∠YXW.

25. Theorems B C E F K L

26. Exit Slip

27. Class Work/Homework
Page 228 #3-17; 19-21, 23, 26, 28

28. Bell Fun: Warm-Up 4.3 Write a congruence statement.
How do you know that ∠N ≅∠R?
Find x.

29. Triangles Congruence by SSS
4.3
I CAN demonstrate that when corresponding sides are congruent the triangles must be congruent.
I CAN prove theorem about triangles.

30. Postulate Side-Side-Side (SSS) Congruence Postulate
If three sides of one triangle are congruent to three sides of a second triangle, then the two triangles are congruent.

32. Examples
Decide whether the congruence statement is true. Explain your reasoning.

34. Example
4. Use the given information to determine if JKL ≅ RST. J(-3, -2), K(0, -2), and L(-3, -8) R(10, 0), S(10, -3), and T(4, 0) y x

35. Class Work/Homework
Page 236 #1-10, 13-19, 24, 26

36. Bell Fun: Warm – Up 4.4

37. Triangle Congruence by SAS & HL
4.4
I CAN demonstrate that when corresponding sides are congruent the triangles must be congruent.
I CAN prove theorems about triangles.
I CAN apply theorems, postulates, or definitions to prove theorems about triangles.

38. Included angle – two sides forming an angle

39. Postulate Side-Angle-Side (SAS) Congruence Postulate
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. Example 1

41. Right Triangle
Legs – the sides adjacent to the right angle; the sides forming the right angle
Hypotenuse – the side opposite of the right angle

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

43. Example 2

44. Examples
Name the congruent triangles. Explain.

45. Closing Question:
What do you need to prove that triangles are congruent?

46. Exit Slip

47. Class Work/Homework
Page 243 #1-14, 16-17, 20-22, 25-27, 32 – 38 even

48. Bell Fun: Warm-Up 4.5
Tell whether the pair of triangles are congruent or not and why.

49. Triangle Congruence by ASA & AAS
4.5
I CAN demonstrate that when corresponding sides and angles are congruent the triangle must be congruent.
I CAN list the sufficient condition to prove triangles are congruent.
I CAN prove theorems about triangles.
I CAN apply theorems, postulates, or definitions to prove theorems about triangles.

50. Included side – the side connecting vertices of two angles

51. Postulates Angle-Side-Angle (ASA) Congruence Postulate
If two angles and the included side of one triangle are congruent to two angles and the included side of a second triangle, then the two triangles are congruent.

52. Theorem Angle-Angle-Side (AAS) Congruence Theorem
If two angles and a non-included side of one triangle are congruent to two angles and the corresponding non-included side of a second triangle, then the two triangles are congruent.

53. Example 1
Can the triangles be proven congruent with the information given in the diagram? If so, state the postulate or theorem you would use. b.

54. CPCTC
Corresponding parts of congruent triangles are congruent

55. Exit Slip

56. Class Work/Homework
Page 252 #3-20, 32, 34

57. Bell Fun: Warm-Up 4.7 Classify each triangle by its sides.
2 cm, 2 cm, 2 cm
7 ft, 11 ft, 7 ft
9 m, 8 m, 10 m
In ABC, if m∠A = 70⁰ and m∠B = 50⁰, what is m∠C?
In DEF, if m∠D = m∠E and m∠F = 26⁰, what are the measures of ∠D and ∠E?

58. Isosceles & Equilateral Triangles
4.7
I CAN use theorems about isosceles and equilateral triangles.

59. Vertex Angle
Isosceles Triangles
leg
leg
Base angles
base
Isosceles Triangle – a triangle where at least two sides are congruent
Legs of An Isosceles Triangle – two congruent sides
Vertex Angle – the angle formed by the legs
Base of Isosceles Triangle – the third side of the triangle; the noncongruent side
Base Angles – two angles adjacent to the base

60. Theorems

61. Examples
Copy and complete the statement.

62. Examples
3. In PQR, PQ ≅ QR. Name two congruent angles.

63. Examples
4. Find the measures of ∠X and ∠Z.

64. Corollaries

65. Examples
5. Find ST in the triangle at the right. 6. Is it possible for an equilateral triangle to have an angle measure other than 60⁰? Explain.

66. Examples
7. Find the values of x and y in the diagram.

67. Exit Slip

68. Class Work/Homework
Page 267 #3-20, 24, 38,