Congruent Triangles 4 4 6 2 Congruence Postulates

Congruent Triangles 4 -4 6 -2 Congruence Postulates. Warm Up Lesson Presentation Lesson Quiz Holt. Geometry Mc. Dougal Geometry

4 -4 Congruent Triangles Warm Up 1. Name all sides and angles of ∆FGH. FG, GH, FH, F, G, H 2. What is true about K and L? Why? ; Third s Thm. 3. What does it mean for two segments to be congruent? They have the same length. Holt Mc. Dougal Geometry

4 -4 Congruent Triangles Objectives 1. Use properties of congruent triangles. Prove triangles congruent by using the definition of congruence. 2. Apply SSS and SAS to construct triangles and solve problems. 3. Prove triangles congruent by using SSS and SAS. 4. Apply ASA, AAS, and HL to construct triangles and to solve problems. 5. Prove triangles congruent by using ASA, AAS. Holt Mc. Dougal Geometry

4 -4 Congruent Triangles Vocabulary corresponding angles corresponding sides congruent polygons SSS SAS ASA AAS included angle included side CPCTC Holt Mc. Dougal Geometry

4 -4 Congruent Triangles 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. Holt Mc. Dougal Geometry

4 -4 Congruent Triangles Holt Mc. Dougal Geometry

4 -4 Congruent Triangles Helpful Hint Two vertices that are the endpoints of a side are called consecutive vertices. For example, P and Q are consecutive vertices. Holt Mc. Dougal Geometry

4 -4 Congruent Triangles 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. Holt Mc. Dougal Geometry

4 -4 Congruent Triangles Helpful Hint When you write a statement such as ABC DEF, you are also stating which parts are congruent. Holt Mc. Dougal Geometry

4 -5 Triangle Congruence: SSS and SAS Triangle Congruence Postulates Remember! Adjacent triangles share a side, so you can apply the Reflexive Property to get a pair of congruent parts. Holt Mc. Dougal Geometry

4 -5 Triangle Congruence: SSS and SAS Example 1: Using SSS to Prove Triangle Congruence Use SSS to explain why ∆ABC ∆DBC. It is given that AC DC and that AB DB. By the Reflexive Property of Congruence, BC BC. Therefore ∆ABC ∆DBC by SSS. Holt Mc. Dougal Geometry

4 -5 Triangle Congruence: SSS and SAS An included angle is an angle formed by two adjacent sides of a polygon. B is the included angle between sides AB and BC. Holt Mc. Dougal Geometry

4 -5 Triangle Congruence: SSS and SAS Triangle Congruence Postulates Caution The letters SAS are written in that order because the congruent angles must be between pairs of congruent corresponding sides. Holt Mc. Dougal Geometry

4 -6 Triangle Congruence: ASA, AAS, and HL An included side is the common side of two consecutive angles in a polygon. The following postulate uses the idea of an included side. Holt Mc. Dougal Geometry

4 -6 Triangle Congruence: ASA, AAS, and HL Holt Mc. Dougal Geometry

4 -6 Triangle Congruence: ASA, AAS, and HL You can use the Third Angles Theorem to prove another congruence relationship based on ASA. This theorem is Angle-Side (AAS). Holt Mc. Dougal Geometry

4 -6 Triangle Congruence: ASA, AAS, and HL Holt Mc. Dougal Geometry

4 -6 Triangle Congruence: ASA, AAS, and HL CPCTC is an abbreviation for the phrase “Corresponding Parts of Congruent Triangles are Congruent. ” It can be used as a justification in a proof after you have proven two triangles congruent. Holt Mc. Dougal Geometry

4 -6 Triangle Congruence: ASA, AAS, and HL Remember! SSS, SAS, ASA, AAS, and HL use corresponding parts to prove triangles congruent. CPCTC uses congruent triangles to prove corresponding parts congruent. Holt Mc. Dougal Geometry