Shortcuts for proving triangles congruent and CPCTC

CPCTC

Corresponding parts of congruent triangles are congruent. You must first establish that you have congruent triangles, usually by one of the 4 valid shortcuts--SSS, SAS, ASA, or AAS. Then you can say a pair of corresponding angles or a pair of corresponding sides are congruent by CPCTC.

HL

If the hypotenuses of two right triangles are congruent and a pair of corresponding legs are also congruent, then the two right triangles are congruent. This is a very special case of SSA that actually works for proving right triangles congruent.

SSS

If three pairs of sides in one triangle are congruent to three pairs of sides in another triangle, then the two triangles are congruent.

ASA

If two angles in one triangle are congruent to two corresponding angles in another triangle and the included sides are congruent, then the two triangles are congruent.

AAS

If two angles of one triangle are congruent to the corresponding angles in another triangle and a pair of corresponding non-included sides are also congruent, then the two triangles are congruent.

SAS

If two sides in one triangle are congruent to two corresponding sides in another triangle, and their included angles are congruent, then the two triangles are congruent.

AAA

This postulate does not guarantee that triangles are congruent. It indicates that three angles in one triangle are congruent to three angles in another triangle. This only means that the two triangles are the same shape, but not necessarily the same size.

SSA

This postulate does not guarantee that triangles are congruent. It indicates that two sides of one triangle and a non-included angle are congruent to two corresponding sides and a corresponding non-included angle of another triangle.