1. Congruent Triangles
Geometry Chapter 4

2. When you finish the test, please pick up a set of 9 index cards
When you finish the test, please pick up a set of 9 index cards. Copy the nine theorems and postulates from chapter four onto these index cards – including the drawings with them. Write the name of each theorem or postulate on the back of the card.
see pages: 199, 205, 206, 213, 214, 228, 235

3. Chapter 4 Standards 2.0 Students write geometric proofs.
4.0 Students prove basic theorems involving congruence.
5.0 Students prove that triangles are congruent, and are able to use the concept of corresponding parts of congruent triangles.
6.0 Students know and are able to use the triangle inequality theorem.
12.0 Students find and use measures of sides and of interior and exterior angles of triangles and polygons to classify figures and solve problems.

4. 4-1 Congruent Figures EQ: How do you show in writing how polygons are congruent?
What do you think makes figures congruent?
They have the same size and shape.
If you can slide, flip or turn a shape so that it fits exactly on another shape, then they are congruent.

5. 4-1 Congruent Figures EQ: How do you show in writing how polygons are congruent?
Congruent polygons have congruent corresponding parts – their matching sides and angles.
Matching vertices are corresponding vertices.
When you name congruent polygons, always list corresponding vertices in the same order.

6. 4-1 Congruent Figures EQ: How do you show in writing how polygons are congruent?

7. 4-1 Congruent Figures EQ: How do you show in writing how polygons are congruent?
Two triangles are congruent if they have three pairs of congruent corresponding sides, and three pairs of congruent corresponding angles. Are the following triangles congruent? Justify your answer.

8. 4-1 Congruent Figures EQ: How do you show in writing how polygons are congruent?
Theorem 4-1: If two angles of one triangle are congruent to two angles of another triangle, then the third angles are congruent.

9. 4-1 Congruent Figures EQ: How do you show in writing how polygons are congruent?

10. 4-1 Congruent Figures EQ: How do you show in writing how polygons are congruent?

11. 4-1 Congruent Figures EQ: How do you show in writing how polygons are congruent?

12. 4-1 Congruent Figures EQ: How do you show in writing how polygons are congruent?

13. 4-2 Triangle Congruence by SSS and SAS EQ: Prove that triangles are congruent.
If you can prove that all sides of two triangles are congruent, then you know the triangles are congruent.

14. 4-2 Triangle Congruence by SSS and SAS EQ: Prove that triangles are congruent.

15. 4-2 Triangle Congruence by SSS and SAS EQ: Prove that triangles are congruent.

16. 4-2 Triangle Congruence by SSS and SAS EQ: Prove that triangles are congruent.
The congruent angle must be the INCLUDED angle between the two sides.

17. 4-2 Triangle Congruence by SSS and SAS EQ: Prove that triangles are congruent.

18. 4-2 Triangle Congruence by SSS and SAS EQ: Prove that triangles are congruent.

19. 4-3Triangle Congruence by ASA and AAS EQ: Are triangles congruent when two angles and a side are congruent?
Warm Up:

20. 4-3Triangle Congruence by ASA and AAS EQ: Are triangles congruent when two angles and a side are congruent?
What does the SAS Postulate say about triangle congruency?

21. 4-3Triangle Congruence by ASA and AAS EQ: Are triangles congruent when two angles and a side are congruent?
At your table, choose any two angle measures that add up to less than 120°. (No zeros)
Agree on a segment length between 5 and 20 centimeters.
Each of you: Draw the line segment, then construct the given angles on each end of the segment to form a triangle.
Measure the two remaining sides and compare your answers.

22. 4-3Triangle Congruence by ASA and AAS EQ: Are triangles congruent when two angles and a side are congruent?
What happened?

23. 4-3Triangle Congruence by ASA and AAS EQ: Are triangles congruent when two angles and a side are congruent?
Which triangles are congruent?

24. 4-3Triangle Congruence by ASA and AAS EQ: Are triangles congruent when two angles and a side are congruent?

25. 4-3Triangle Congruence by ASA and AAS EQ: Are triangles congruent when two angles and a side are congruent?

26. 4-3Triangle Congruence by ASA and AAS EQ: Are triangles congruent when two angles and a side are congruent?

27. Retake for Chapter 3 test:
homework:
page 215 (1-15) all
Retake for Chapter 3 test:
Pick up a retake practice packet.
Complete test corrections.
You MUST have all chapter 3 homework completed and come in for at least 1 enrichment period before the retake next Thursday.

29. 4-4: CPCTC EQ: Are all parts of congruent triangles congruent?
warm up

30. 4-4: CPCTC EQ: Are all parts of congruent triangles congruent?
Once you show that triangles are congruent using SSS, SAS, ASA or AAS, then you can make conclusions about the other parts of the triangles because, by definition, congruent parts of congruent triangles are congruent.
Abbreviate this CPCTC

31. 4-4: CPCTC EQ: Are all parts of congruent triangles congruent?
Before you can use CPCTC in a proof, you must first show that the triangles are congruent.

32. 4-4: CPCTC EQ: Are all parts of congruent triangles congruent?

33. 4-4: CPCTC EQ: Are all parts of congruent triangles congruent?

34. 4-4: CPCTC EQ: Are all parts of congruent triangles congruent?

35. 4-4: CPCTC EQ: Are all parts of congruent triangles congruent?

36. 4-5 Isosceles and Equilateral Triangles EQ: How do you use the properties of Isosceles triangles in proofs?
Warm Up

37. 4-5 Isosceles and Equilateral Triangles EQ: How do you use the properties of Isosceles triangles in proofs?
Construct an Isosceles Triangle
1. Use a straight edge to make a line segment. Label the endpoints A and B.
2. Set your compass to a length that is greater than half the length of the segment.
3. Without changing the compass setting, make arcs from either end of the line segment.
4. Connect the endpoints of the segment to the intersection point of the two arcs. Label this point C.
5. Measure the sides of the triangle to confirm that they are equal.

38. Label the point where the fold intersects AB as point D.
4-5 Isosceles and Equilateral Triangles EQ: How do you use the properties of Isosceles triangles in proofs?
Fold your triangle carefully in half, so points A and B are exactly on top of each other.
Label the point where the fold intersects AB as point D.
What appears to be true of angles A and B?
What appears to be true of the intersection of CD and AB?
Write a conjecture about the angles opposite the congruent sides of an isosceles triangle.

39. 4-5 Isosceles and Equilateral Triangles EQ: How do you use the properties of Isosceles triangles in proofs?

40. 4-5 Isosceles and Equilateral Triangles EQ: How do you use the properties of Isosceles triangles in proofs?

41. 4-5 Isosceles and Equilateral Triangles EQ: How do you use the properties of Isosceles triangles in proofs?

42. 4-5 Isosceles and Equilateral Triangles EQ: How do you use the properties of Isosceles triangles in proofs?

43. A corollary is a statement that follows directly from a theorem
4-5 Isosceles and Equilateral Triangles EQ: How do you use the properties of Isosceles triangles in proofs?
A corollary is a statement that follows directly from a theorem

44. 4-5 Isosceles and Equilateral Triangles EQ: How do you use the properties of Isosceles triangles in proofs?

45. 4-5 Isosceles and Equilateral Triangles EQ: How do you use the properties of Isosceles triangles in proofs?

46. 4-5 Isosceles and Equilateral Triangles EQ: How do you use the properties of Isosceles triangles in proofs?

47. 4-6 Congruence in Right Triangles EQ: What are the theorems about right triangles?
Warm Up

48. 4-6 Congruence in Right Triangles EQ: What are the theorems about right triangles?

49. 4-6 Congruence in Right Triangles EQ: What are the theorems about right triangles?

50. 4-6 Congruence in Right Triangles EQ: What are the theorems about right triangles?

51. 4-6 Congruence in Right Triangles EQ: What are the theorems about right triangles?

52. 4-6 Congruence in Right Triangles EQ: What are the theorems about right triangles?
Homework: p237 (1-8)

53. 4-7 Using Corresponding Parts of Congruent Triangles
Warm Up:

54. 4-7 Using Corresponding Parts of Congruent Triangles
When a geometric drawing is complicated, it is sometimes helpful to separate it into more than one drawing.

55. 4-7 Using Corresponding Parts of Congruent Triangles

56. 4-7 Using Corresponding Parts of Congruent Triangles

57. 4-7 Using Corresponding Parts of Congruent Triangles
Sometimes you can prove triangles are congruent and then use their corresponding parts to prove another pair congruent.

58. 4-7 Using Corresponding Parts of Congruent Triangles

59. 4-7 Using Corresponding Parts of Congruent Triangles
Worksheet 4-7, both sides
Chapter 4 test Tuesday – period 3
Chapter 4 test Wednesday – period 6

60. Chapter 4 Review Questions
Draw RSTU congruent to GHIJ.
List all the congruent parts of the two figures. h i j g

61. Chapter 4 Review Questions
What else would you need to have to prove these triangles congruent by SSS?
By SAS?

62. Chapter 4 Review Questions
What other piece of information do you need to prove these triangles are congruent?
By ASA?
By SAS?
By AAS?

63. Chapter 4 Review Questions
prove ∠P ≅∠Q

64. Chapter 4 Review Questions
prove ∠P ≅∠Q

65. Chapter 4 Review Questions
prove ∠P ≅∠Q

66. Chapter 4 Review Questions

67. Chapter 4 Review Questions

68. Chapter 4 Review Questions
Given: KM≅LJ, KJ ≅ LM
Prove: OJ ≅ OM