4.7 ASA and AAS Objectives: Apply ASA and AAS to construct triangles and to solve problems. Prove triangles congruent by using ASA and AAS.

Warm up Given DEF and GHI, if D  G and E  H, why is F  I? Third s Thm. If two angles in one triangle are congruent to two angles in another triangle, then the third pair of angles must also congruent. This is called the Third Angle Theorem. If ∠A≅∠D and ∠B≅∠E, then ∠C≅∠F.

Can someone explain what the “included” angle means in SAS? So explain what the “included” side means in ASA? An included side is the common side of two consecutive angles in a triangle. The following theorem uses the idea of an included side.

R and K 2. X and R 3. 8 and 9 4. 10 and 12 Name the included side between each pair of angles. R and K 2. X and R 3. 8 and 9 4. 10 and 12 5. 5 and 1 6. 4 and 2 RK RX CX XD SR QR

Example 1: Applying ASA Congruence Determine if you can use ASA to prove the triangles congruent. Explain. Two congruent angle pairs are given, but the included sides are not given as congruent. So, we cannot use ASA.

Check It Out! Example 2 Determine if you can use ASA to prove NKL  LMN. Explain. 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.

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

State the theorem that you would use to prove the triangles congruent State the theorem that you would use to prove the triangles congruent. Name the congruent triangles. Neither SSS AAS SAS

State the theorem that you would use to prove the triangles congruent State the theorem that you would use to prove the triangles congruent. Name the congruent triangles. SAS ASA

A A S S A A Check It Out! Example 3 Prove the triangles congruent. Given: JL bisects KLM, K  M Prove: JKL  JML A A S S A A

USE: AAS A A S 1. Given 2. Given 3. def angle bisector ✔ ✔ ✔ USE: AAS Statements Reasons 1. Given A 2. Given 3. def angle bisector A 4. Reflexive Property S 5. AAS