4.5 Proving Triangles Congruent - ASA and AAS *Then: You proved triangles congruent using SSS and SAS *Now: 1. Use the ASA Postulate to test for congruence. 2. Use the AAS Theorem to test for congruence.

Review SSS SAS Two ways to prove triangles are congruent- http://www.cliffsnotes.com/study_guide/Congruent-Triangles.topicArticleId-18851,articleId-18788.html

Postulate 4.3: Angle-Side-Angle (ASA) Congruence 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. If Angle  A  D, Side AC  DF, and Angle  C  F, then  ABC  DEF.

Example 1: Write a two-column proof Given: AB  CD ACD  CAB Prove: ABC   CDA Statements Reasons 1. AB  CD 1. __________________ 2. ACD  CAB 2. __________________ 3. BCA  DAC 3. __________________ 4. BD  BD 4. __________________ 5. ABC   CDA 5. __________________

Example 2: Write a flow proof Given: S  V T is the midpoint of SV Prove:  RTS  UTV S  V T is the midpoint of SV RTS  VTU ST  TV  RTS  UTV

Example 3: Write a paragraph proof Given: CD bisects AE, AB  CD E  BCA Prove:  ABC  CDE It is _________ that ____ _____ and CD _____ AE. So, ____  _____ by definition of ____________. Also given that ___  ___, so ____ _____ by __________________________________________. Then  ____  _____ by _____________________

Theorem 4.5: Angle-Angle-Side (AAS) Congruence If two angles and a nonincluded side of one triangle are congruent to two angles and the corresponding nonincluded side of a second triangle, then the two triangles are congruent. If Angle  A  D, Angle  C  F, and Side BC  EF, then  ABC  DEF.

Example 4: Complete the proof. Given:  G  B, CB  GA Prove: GCA  BAC Statements Reasons 1. CB  GA 1. ___________________ 2.  BCA  GAC 2. ___________________ 3.  G  B 3. ___________________ 4. AC  AC 4. ___________________ 5. GCA  BAC 5. ___________________

Example 5 : Write a flow proof ADB  ACE EC  DB AEC  ABD AEC  ABD A  A

Example 6: Write a paragraph proof Given:  S  U TR bisects STU Prove: SRT  URT It is given that ______bisects ______ so  _____  ______ by ______________________. Also given is ________________. _____  _____ by __________________________________________. Then _____  ______ by _______________________. Finally,  _____  ______ by ___________________ .

4.5 Assignment: p. 279-282- See handout