Proving Triangles Congruent

The Idea of a Congruence Two geometric figures with exactly the same size and shape. A C B D E F

How much do you need to know. . . . . . about two triangles to prove that they are congruent?

Corresponding Parts ABC   DEF AB  DE BC  EF AC  DF  A   D  B   E  C   F B A C ABC   DEF E D F

Do you need all six ? NO ! SSS SAS ASA AAS

Side-Side-Side (SSS) If three sides of one triangle are congruent to three sides of a second triangle, then the two triangles are congruent.

Side-Side-Side (SSS) AB  DE BC  EF AC  DF ABC   DEF B A C E D F

Side-Angle-Side (SAS) If two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then the two triangles are congruent.

Side-Angle-Side (SAS) B E F A C D AB  DE A   D AC  DF ABC   DEF included angle

Included Side The side between two angles GI GH HI

Angle-Side-Angle (ASA) 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.

Angle-Side-Angle (ASA) B E F A C D A   D AB  DE  B   E ABC   DEF included side

Angle-Angle-Side (AAS) 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.

Angle-Angle-Side (AAS) B E F A C D A   D  B   E BC  EF ABC   DEF Non-included side

There is no such thing as an SSA postulate! Warning: No SSA Postulate There is no such thing as an SSA postulate! E B F A C D NOT CONGRUENT

There is no such thing as an AAA postulate! Warning: No AAA Postulate There is no such thing as an AAA postulate! E B A C F D NOT CONGRUENT

The Congruence Postulates SSS correspondence ASA correspondence SAS correspondence AAS correspondence SSA correspondence AAA correspondence

Name That Postulate (when possible) SAS ASA SSA SSS

Name That Postulate SAS SAS SSA SAS Vertical Angles Reflexive Property (when possible) Vertical Angles Reflexive Property SAS SAS Vertical Angles Reflexive Property SSA SAS