Theorems for Proving Triangle Congruence

There are several theorems and postulates that can be used to prove that triangles are congruent. The term "congruent" means that two triangles have the same shape and size. When proving triangle congruence, you typically need to show that all corresponding angles and sides of the triangles are equal. Here are some of the most commonly used theorems and postulates for proving triangle congruence:

1. Side-Angle-Side (SAS) Theorem:

   • 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.

2. Angle-Side-Angle (ASA) Theorem:

   • 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.

3. Side-Side-Side (SSS) Theorem:

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

4. Angle-Angle-Side (AAS) Theorem:

   • If two angles and a non-included side of one triangle are congruent to two angles and a non-included side of another triangle, then the two triangles are congruent.

5. Hypotenuse-Leg (HL) Theorem (for right triangles):

   • If the hypotenuse and one leg of one right triangle are congruent to the hypotenuse and one leg of another right triangle, then the two right triangles are congruent.

6. Corresponding Parts of Congruent Triangles Are Congruent (CPCTC):

   • Once you have proven that two triangles are congruent using one of the above theorems, you can conclude that all corresponding angles and sides of the two triangles are congruent.

These theorems and postulates are used in geometry to demonstrate that two triangles are identical in size and shape. When applying them, it's important to provide sufficient evidence and logical reasoning to support your claims.