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 you learned that if all six pairs of corresponding parts (sides and angles) are congruent, then the triangles are congruent. B A C AB  DE BC  EF AC  DF  A   D  B   E  C   F ABC   DEF E D F

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

Side-Side-Side (SSS) AB  DE BC  EF AC  DF ABC   DEF If 3 sides of one triangle are congruent to 3 sides of another triangle, then the triangles are congruent

Side-Angle-Side (SAS) B E F A C D AB  DE A   D AC  DF ABC   DEF included side If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent

Included Angle The angle between two sides  H  G  I

Included Angle Name the included angle: YE and ES ES and YS YS and YE

Angle-Side-Angle (ASA) B E F A C D A   D AB  DE  B   E ABC   DEF included side If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent.

Included Side The side between two angles GI GH HI

Included Side Name the included angle: Y and E E and S S and Y YE ES SY

Angle-Angle-Side (AAS) B E F A C D A   D  B   E BC  EF ABC   DEF Non-included side If 2 angles and a non-included side of 1 triangle are congruent to 2 angles and the corresponding non-included side of another triangle, then the 2 triangles are congruent

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 (when possible) AAA ASA SSA SAS

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

HW: Name That Postulate (when possible)

HW: Name That Postulate (when possible)

Let’s Practice B  D AC  FE A  F Indicate the additional information needed to enable us to apply the specified congruence postulate. For ASA: B  D For SAS: AC  FE A  F For AAS:

Indicate the additional information needed to enable us to apply the specified congruence postulate. For ASA: For SAS: For AAS: