Triangle Congruence 5390C4

1. **Problem Statement:** We want to describe the five ways to prove that two triangles are congruent. 2. **The Five Congruence Theorems:** - **SSS (Side-Side-Side):** If three sides of one triangle are equal to three sides of another triangle, the triangles are congruent. - **SAS (Side-Angle-Side):** If two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, the triangles are congruent. - **ASA (Angle-Side-Angle):** If two angles and the included side of one triangle are equal to two angles and the included side of another triangle, the triangles are congruent. - **AAS (Angle-Angle-Side):** If two angles and a non-included side of one triangle are equal to two angles and the corresponding non-included side of another triangle, the triangles are congruent. - **HL (Hypotenuse-Leg) for right triangles:** If the hypotenuse and one leg of a right triangle are equal to the hypotenuse and one leg of another right triangle, the triangles are congruent. 3. **Sketch and Definitions:** - Each theorem involves comparing specific parts of two triangles. - Congruent triangles have all corresponding sides and angles equal. 4. **Examples:** - **SSS Example:** Triangles with sides $3, 4, 5$ and $3, 4, 5$ are congruent by SSS. - **SAS Example:** Triangles with sides $5, 7$ and included angle $60^\circ$ and another triangle with the same measurements are congruent by SAS. - **ASA Example:** Triangles with angles $50^\circ, 60^\circ$ and included side $8$ and another triangle with the same are congruent by ASA. - **AAS Example:** Triangles with angles $40^\circ, 70^\circ$ and side $6$ (not between the angles) and another triangle with the same are congruent by AAS. - **HL Example:** Right triangles with hypotenuse $10$ and leg $6$ and another right triangle with the same are congruent by HL. 5. **Summary:** These five theorems provide reliable methods to prove triangle congruence by comparing sides and angles in specific ways.