5.6 Proving Triangles Congruent using ASA and AAS Geometry EQ: What other ways can you prove triangles congruent?

Geometry 5.6 Proving Triangles Congruent: ASA, AAS Goals Use ASA and AAS postulates to prove two triangles are congruent. Solve problems using congruence postulates. June 10, 2018 Geometry 5.6 Proving Triangles Congruent: ASA, AAS

Geometry 5.6 Proving Triangles Congruent: ASA, AAS Included Sides AC is included between A and C. C AB is included between A and B. CB is included between C and B. B A June 10, 2018 Geometry 5.6 Proving Triangles Congruent: ASA, AAS

Theorem 5.10: Angle-Side-Angle If two angles and the included side of one triangle are congruent to corresponding parts of another triangle, then the triangles are congruent. (ASA) A B C X Y Z ABC  XYZ June 10, 2018 Geometry 5.6 Proving Triangles Congruent: ASA, AAS

Example Prove ADB  CBD Proof: ADB  CBD (Given) ABD  CDB (Given) ADB  CBD (ASA) A S DB  DB (Ref.) A A B June 10, 2018 Geometry 5.6 Proving Triangles Congruent: ASA, AAS

Theorem 5.11 Angle-Angle-Side If two angles and a nonincluded side of one triangle are congruent to corresponding parts of another triangle, then the triangles are congruent. A B C D E F ABC  DEF June 10, 2018 Geometry 5.6 Proving Triangles Congruent: ASA, AAS

Geometry 5.6 Proving Triangles Congruent: ASA, AAS Example Is ABC  ADC? A B C D Yes,by AAS S A A June 10, 2018 Geometry 5.6 Proving Triangles Congruent: ASA, AAS

Are these triangles congruent? Yes. Reason: AAS OR AAS. June 10, 2018 Geometry 5.6 Proving Triangles Congruent: ASA, AAS

Are these triangles congruent? You could even use ASA. June 10, 2018 Geometry 5.6 Proving Triangles Congruent: ASA, AAS

Overlapping Triangles C A D A B D C A D A B D June 10, 2018 Geometry 5.6 Proving Triangles Congruent: ASA, AAS

Geometry 5.6 Proving Triangles Congruent: ASA, AAS Problem: Is ABD  DCA? C B 85° 85° 15° 15° D A YES by AAS June 10, 2018 Geometry 5.6 Proving Triangles Congruent: ASA, AAS

Geometry 5.6 Proving Triangles Congruent: ASA, AAS Problem: Is ABD  DCA? C C 85° 15° D B 85° A B 15° 85° 85° 15° 15° D A YES by AAS June 10, 2018 Geometry 5.6 Proving Triangles Congruent: ASA, AAS

Methods to show triangles congruent: SSS SAS ASA AAS When trying to determine if triangles are congruent, carefully study the given information, mark the triangles if necessary, then determine what pattern you have. June 10, 2018 Geometry 5.6 Proving Triangles Congruent: ASA, AAS

Geometry 5.6 Proving Triangles Congruent: ASA, AAS Problem A S S A B S A C R A A What other congruence must be given to show that ABC  RST using the AAS postulate? BC  ST June 10, 2018 Geometry 5.6 Proving Triangles Congruent: ASA, AAS

Geometry 5.6 Proving Triangles Congruent: ASA, AAS Problem Alternate Solution A S B S A A C R A A S What other congruence must be given to show that ABC  RST using the AAS postulate? BA  SR June 10, 2018 Geometry 5.6 Proving Triangles Congruent: ASA, AAS

Geometry 5.6 Proving Triangles Congruent: ASA, AAS Problem A B S A C S A R A S A What other congruence must be given to show that ABC  RST using the ASA postulate? AC  RT June 10, 2018 Geometry 5.6 Proving Triangles Congruent: ASA, AAS

Geometry 5.6 Proving Triangles Congruent: ASA, AAS State the third congruence that must be given to prove the given triangles congruent using the indicated postulate or theorem. June 10, 2018 Geometry 5.6 Proving Triangles Congruent: ASA, AAS

Geometry 5.6 Proving Triangles Congruent: ASA, AAS 1. ASA Theorem. C A B F D E C  F June 10, 2018 Geometry 5.6 Proving Triangles Congruent: ASA, AAS

Geometry 5.6 Proving Triangles Congruent: ASA, AAS 2. AAS Theorem. A B C E D F C  F June 10, 2018 Geometry 5.6 Proving Triangles Congruent: ASA, AAS

Geometry 5.6 Proving Triangles Congruent: ASA, AAS 3. SSS A C B E F D CB  FE June 10, 2018 Geometry 5.6 Proving Triangles Congruent: ASA, AAS

Geometry 5.6 Proving Triangles Congruent: ASA, AAS 4. SAS Theorem. F E A D C B A  D June 10, 2018 Geometry 5.6 Proving Triangles Congruent: ASA, AAS

Geometry 5.6 Proving Triangles Congruent: ASA, AAS Is it possible to prove that the triangles are congruent? If so, state the postulate or theorem you would use. Explain reasoning June 10, 2018 Geometry 5.6 Proving Triangles Congruent: ASA, AAS

Geometry 5.6 Proving Triangles Congruent: ASA, AAS X Z W Yes AAS June 10, 2018 Geometry 5.6 Proving Triangles Congruent: ASA, AAS

Geometry 5.6 Proving Triangles Congruent: ASA, AAS 6. C A Yes SAS D R June 10, 2018 Geometry 5.6 Proving Triangles Congruent: ASA, AAS

Geometry 5.6 Proving Triangles Congruent: ASA, AAS 7. The only other thing we know for sure: SOT  LOF (Vertical Angles) But this spells… SSA NOT CONGRUENT S T O F L June 10, 2018 Geometry 5.6 Proving Triangles Congruent: ASA, AAS

Geometry 5.6 Proving Triangles Congruent: ASA, AAS Given: LA || SN, LR  NR Prove: LAR  NSR 1. LA || SN 1. Given 2. L  N 2. Alt. int s 3. LR  NR 3. Given 4. LRA  NRS 4. Vert s 5. LAR  NSR 5. ASA June 10, 2018 Geometry 5.6 Proving Triangles Congruent: ASA, AAS

List four ways to show triangles are congruent. SSS ASA SAS AAS List one way that won’t work. SSA June 10, 2018 Geometry 5.6 Proving Triangles Congruent: ASA, AAS

Geometry 5.6 Proving Triangles Congruent: ASA, AAS Important! To show two triangles congruent, at least one side must be known. Three angles (AAA) does not show congruence (but it is still useful). June 10, 2018 Geometry 5.6 Proving Triangles Congruent: ASA, AAS

Geometry 5.6 Proving Triangles Congruent: ASA, AAS Summarize June 10, 2018 Geometry 5.6 Proving Triangles Congruent: ASA, AAS