Proving Triangles Congruent: Methods and Examples

Explore different methods such as ASA and AAS postulates and theorems to prove triangles congruent. Detailed examples and proofs provided for better understanding.

• Triangles
• Congruence
• Proofs
• Geometry
• Postulates

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1. SECTION 4.4 MORE WAYS TO PROVE TRIANGLES CONGRUENT

2. POSTULATE 19 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. C F E B D A <A = <D; AC = DF; <C =<F; ABC DEF

3. THEOREM 4.7 Angle-Angle-Side (AAS) Congruence Theorem If two angles and a non-included side one triangle are congruent to two angles and the corresponding non-included side of a second triangle, then the two triangles are congruent. F C E B D A <A <D, <B <E, BC EF then ABC DEF

4. EXAMPLE 1: Is it possible to prove these triangles are congruent? If so, state the postulate or theorem you would use. Explain your reasoning.

5. EXAMPLE 2: Is it possible to prove these triangles are congruent? If so, state the postulate or theorem you would use. Explain your reasoning.

6. EXAMPLE 2: In addition to the congruent segments that are marked, NP NP. Two pairs of corresponding sides are congruent. This is not enough information to prove the triangles are congruent.

7. PROVE Given: KL LA, KJ JA, AK bisects <LAJ Prove: ??? ??? L K A J

8. EX. 2 PROVING TRIANGLES ARE CONGRUENT Given: AD EC, BD BC Prove: ABD EBC C A Plan for proof: Notice that ABD and EBC are congruent. You are given that BD BC B E D . Use the fact that AD EC to identify a pair of congruent angles.

9. C A B PROOF: E D Statements: 1. BD BC 2. AD EC 3. D C 4. ABD EBC 5. ABD EBC Reasons: 1.

10. C A B PROOF: E D Statements: 1. BD BC 2. AD EC 3. D C 4. ABD EBC 5. ABD EBC Reasons: 1. Given 2. Given 3. Alternate Interior Angles 4. Vertical Angles Theorem