4-3

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

2. FG, GH, FH, F, G, H Warm Up 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?  ;Third s Thm. They have the same length.

3. Objectives Use properties of congruent triangles. Prove triangles congruent by using the definition of congruence.

4. November 8, 2013 Congruent Triangles What are the properties of congruent triangles using the definition of congruence?

5. Vocabulary corresponding angles corresponding sides congruent polygons

6. 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.

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

9. 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.

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

11. Sides: PQ ST, QR  TW, PR  SW Example 1: Naming Congruent Corresponding Parts Given: ∆PQR ∆STW Identify all pairs of corresponding congruent parts. Angles: P  S, Q  T, R  W

12. Sides: LM EF, MN  FG, NP  GH, LP  EH Check It Out! Example 1 If polygon LMNP polygon EFGH, identify all pairs of corresponding congruent parts. Angles: L  E, M  F, N  G, P  H

13. For example, you only need to know that two triangles have three pairs of congruent corresponding sides. This can be expressed as the following postulate.

14. Remember! Adjacent triangles share a side, so you can apply the Reflexive Property to get a pair of congruent parts.

15. It is given that AC DC and that AB  DB. By the Reflexive Property of Congruence, BC  BC. Therefore ∆ABC  ∆DBC by SSS. Example 2: Using SSS to Prove Triangle Congruence Use SSS to explain why ∆ABC  ∆DBC.

16. It is given that AB CD and BC  DA. By the Reflexive Property of Congruence, AC  CA. So ∆ABC  ∆CDA by SSS. On your TABLE! Example 2 Use SSS to explain why ∆ABC  ∆CDA.

17. An included angle is an angle formed by two adjacent sides of a polygon. B is the included angle between sides AB and BC.

18. It can also be shown that only two pairs of congruent corresponding sides are needed to prove the congruence of two triangles if the included angles are also congruent.

20. Caution The letters SAS are written in that order because the congruent angles must be between pairs of congruent corresponding sides.

21. It is given that XZ VZ and that YZ  WZ. By the Vertical s Theorem. XZY  VZW. Therefore ∆XYZ  ∆VWZ by SAS. Example 3: Engineering Application Use SAS to explain why ∆XYZ  ∆VWZ.

22. It is given that BA BD and ABC  DBC. By the Reflexive Property of , BC  BC. So ∆ABC  ∆DBC by SAS. Check It Out! Example 3 Use SAS to explain why ∆ABC  ∆DBC.

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31. Definitions

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45. An included side is the common side of two consecutive angles in a polygon. The following postulate uses the idea of an included side.

47. Example 4: Applying ASA Congruence Determine if you can use ASA to prove the triangles congruent. Explain. Two congruent angle pairs are give, but the included sides are not given as congruent. Therefore ASA cannot be used to prove the triangles congruent.

48. By the Alternate Interior Angles Theorem. KLN  MNL. NL  LN by the Reflexive Property. No other congruence relationships can be determined, so ASA cannot be applied. Check It Out! Example 4 Determine if you can use ASA to prove NKL  LMN. Explain.

49. You can use the Third Angles Theorem to prove another congruence relationship based on ASA. This theorem is Angle-Angle-Side (AAS).