Ways to prove Triangles Congruent

Method: Side-Side-Side (SSS) Description: Three corresponding sides are congruent from one triangle to another. (SSS is a Postulate)

Method: Side-Angle-Side (SAS) Description: Two corresponding sides and the included angle are congruent to the corresponding parts of another triangle.

Method: Angle-Side-Angle (ASA) Description: Two angles and the included side are congruent from one triangle to another triangle.

Method: Angle-Angle-Side (AAS) Description: By using ASA, we know the third angle in one triangle must be congruent to the third angle in the other triangle since the angle sum must be 180 degrees. (AAS is a Theorem and is a shortcut of the ASA Postulate)

Ways to Prove Right Triangles Congruent *Keep in mind each of these methods only contain 2 letters because the right angle of the triangle is always a given so it is not included as one of the letters.

Method: Leg-Leg (LL) Description: Same as SAS but the “A” is the right angle and so you are left with just two sides or “legs” of the right triangle.

Method: Leg-Angle (LA) Description: Same as ASA or AAS but one of the angles is the right angle so you are left with just one angle and side or an angle and a leg. Like ASA or Like AAS

Method: Hypotenuse-Angle (HA) Description: Same as AAS but one of the angles is the right angle the other angle is one of the acute angles and the side is the hypotenuse of the triangle.

Method: Hypotenuse-Leg (HL) Description: This is really Angle-Side- Side (can’t use…spells bad word), but it only works when the angle is a right angle, so you are left with the sides being a leg and hypotenuse of the right triangle.

Ways that you CANNOT prove triangles congruent

Method: Angle-Side-Side (ASS) Reason cannot be proven: If you draw two triangles with one congruent angle and one congruent side next to the angle, the next side can still be the same length on both triangles, but it can move to change the measure of the angle…therefore the triangles would not be congruent. *There is one exception…this is when the angle is a right angle as mentioned above.

Method: Angle-Angle-Angle (AAA) Reason cannot be proven: Just because the three angles are congruent in two triangles does not necessarily mean that they are congruent, because they could be different sizes which would make them similar.