4-3

1. 4-3 Congruent Triangles Holt Geometry Warm Up Lesson Presentation Lesson Quiz

2. Do Now 1.Name all sides and angles of ∆FGH. 2. What is true about K and L? Why? 3.What does it mean for two segments to be congruent?

3. Objectives TSW use properties of congruent triangles. TSW prove triangles congruent by using the definition of congruence.

4. Vocabulary corresponding angles corresponding sides congruent polygons

5. Geometric figures are congruent if they are the same size and shape. Corresponding angles and corresponding sides are in the same position in polygons with an equal number of sides. Two polygons are congruent polygons if and only if their corresponding sides are congruent. Thus triangles that are the same size and shape are congruent.

7. Helpful Hint Two vertices that are the endpoints of a side are called consecutive vertices. For example, P and Q are consecutive vertices.

8. To name a polygon, write the vertices in consecutive order. For example, you can name polygon PQRS as QRSP or SRQP, but not as PRQS. In a congruence statement, the order of the vertices indicates the corresponding parts.

9. Helpful Hint When you write a statement such as ABCDEF, you are also stating which parts are congruent.

10. Example 1: Naming Congruent Corresponding Parts Given: ∆PQR ∆STW Identify all pairs of corresponding congruent parts.

11. Example 2 If polygon LMNP polygon EFGH, identify all pairs of corresponding congruent parts.

12. Example 3: Using Corresponding Parts of Congruent Triangles Given: ∆ABC ∆DBC. Find the value of x.

13. Example 3: Using Corresponding Parts of Congruent Triangles Given: ∆ABC ∆DBC. Find mDBC.

14. Example 4 Given: ∆ABC  ∆DEF Find the value of x.

15. Example 5 Given: ∆ABC  ∆DEF Find mF.

16. Given:YWXandYWZ are right angles. YW bisects XYZ. W is the midpoint of XZ. XY  YZ. Prove: ∆XYW  ∆ZYW Example 6: Proving Triangles Congruent

17. 5. 1.YWX and YWZ are rt. s. 1. Given 2.YW bisects XYZ 2. Given 3.W is mdpt. of XZ 3. Given 4. 4. 5. 6. 6. 7. 7 8. 8. 9. 9. 10.∆XYW  ∆ZYW 10.

18. Example 7 Given:ADbisectsBE. BEbisectsAD. ABDE, A D Prove:∆ABC  ∆DEC

19. 1. 1. Given 2. 2. Given 3. 3. Given 4. 4. Given 5. 5. 6. 6. 7. 7 8. 8.

20. Example 8: Engineering Application The diagonal bars across a gate give it support. Since the angle measures and the lengths of the corresponding sides are the same, the triangles are congruent. Given: PR and QT bisect each other. PQS  RTS, QP  RT Prove: ∆QPS ∆TRS

21. 1. 1. Given 2. 2. Given 3. 3. Given 4. 4. 5. 5. 6. 6. 7. 7 8. 8. 9. 9.

22. Given: MK bisects JL. JL bisects MK. JK ML.JK|| ML. Example 9 Use the diagram to prove the following. Prove: ∆JKN ∆LMN

23. 1. 1. Given 2. 2. Given 3. 3. Given 4. 4. 5. 5. 6. 6. 7. 7 8. 8.

24. 10. ∆ABC  ∆JKL and AB = 2x + 12. JK = 4x – 50. Find x and AB.

25. 11. Given that polygon MNOP polygon QRST, identify the congruent corresponding part. NO  ____ T  ____

26. 12. Given: C is the midpoint of BD and AE. A  E, AB  ED Prove: ∆ABC  ∆EDC

27. Lesson Quiz 1. ∆ABC  ∆JKL and AB = 2x + 12. JK = 4x – 50. Find x and AB. Given that polygon MNOP polygon QRST, identify the congruent corresponding part. 2. NO  ____ 3. T  ____ 4. Given: C is the midpoint of BD and AE. A  E, AB  ED Prove: ∆ABC  ∆EDC

28. Statements Reasons 1. 1. 2. 2. 3. 3. 4. AC EC; BC  DC 4. 5. ACB  ECD 5. 6. 6. Third s Thm. 7. 7. Lesson Quiz 4.

29. RS Lesson Quiz 1. ∆ABC  ∆JKL and AB = 2x + 12. JK = 4x – 50. Find x and AB. Given that polygon MNOP polygon QRST, identify the congruent corresponding part. 2. NO  ____ 3. T  ____ 4. Given: C is the midpoint of BD and AE. A  E, AB  ED Prove: ∆ABC  ∆EDC 31, 74 P

30. Statements Reasons 1. A  E 1. Given 2. C is mdpt. of BD and AE 2. Given 3. AC EC; BC  DC 3. Def. of mdpt. 4. AB ED 4. Given 5. ACB  ECD 5. Vert. s Thm. 6. B  D 6. Third s Thm. 7. ABC  EDC 7. Def. of  ∆s Lesson Quiz 4.