SAS SSS SAS SSS

Sect. 4.4 Proving Triangles are Congruent: ASA and AAS Goal 1 Using the ASA and AAS Congruence Methods Goal 2 Using Congruence Postulate and Theorems

Postulate 21 Angle-Side-Angle (ASA) Congruence Postulate Using the ASA and AAS Congruence Methods Postulate 21 Angle-Side-Angle (ASA) Congruence Postulate If, in two triangles, two angles and the included side of one triangle are congruent to two angles and the included side of the other, then the triangles are congruent.                                                                                                                    

Using the ASA and AAS Congruence Methods Given: ABC   DCB;  DBC   ACB Prove: ABC  DCB Statements Reasons 1. ABC   DCB; DBC  ACB 1. Given 2. Reflexive 3. ABC  DCB 3. ASA 2.

Theorem 4.5 Angle-Angle-Side (AAS) Congruence Theorem Using the ASA and AAS Congruence Methods Theorem 4.5 Angle-Angle-Side (AAS) Congruence Theorem If, in two triangles, two angles and a non-included side of one triangle are congruent respectively to two angles and the corresponding non-included side of the other, then the triangles are congruent.

Using the ASA and AAS Congruence Methods Given: B  C; D  F; M is the midpoint of Prove: BDM  CFM Statements Reasons 1. B  C; D  F; M is the midpoint of 1. Given 2. 2. Definition of Midpoint 3. AAS 3. BDM  CFM

Using the ASA and AAS Congruence Methods Given: bisects XZY and XWY Prove: WZX  WZY Statements Reasons 1. bisects XZY and XWY 1. Given 2. XZW  YZW; XWZ  YWZ 2. Definition of Angle Bisector 3. 3. Reflexive 4. WZX  WZY 4. ASA

Methods of Proving Triangles Congruent Using Congruence Postulates and Theorems Methods of Proving Triangles Congruent SSS If three sides of one triangle are congruent to three sides of another triangle, the triangles are congruent. SAS If two sides and the included angle of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent. ASA If two angles and the included side of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent. AAS If two angles and the non-included side of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent.

Using Congruence Postulates and Theorems AAA works fine to show that triangles are the same SHAPE (similar), but does NOT work to show congruent! You can draw 2 equilateral triangles that are the same shape but not the same size.

What about two sides and a not-included angle? Using Congruence Postulates and Theorems What about two sides and a not-included angle? Note that GB and BH are the same length, and that AB and angle A are the other  parts of Angle – Side – Side.

Using Congruence Postulates and Theorems ASS It does NOT work!!!

Given: CB  AD ; CB bisects ACD Prove: ABC  DBC Using Congruence Postulates and Theorems Given: CB   AD ; CB bisects ACD           Prove: ABC   DBC

Given: A  E ; B  G ; AC  EF Prove: ABC  EGF Baby Proof

A Given: BAC  DAE ; B  D ; AC  AE Prove: ABC  ADE B E C D

Given: 2  3 ; B  D ; AC  AE Prove: ABE  ADC

Given: ○A ; 2  3 ; B  D Prove: ABE  ADC

Homework pp. 223 Exs. 8 – 23, 34-38