1. **Problem Statement:** Prove the SAS (Side-Angle-Side) congruence theorem which states that if two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, then the triangles are congruent. 2. **Setup:** Consider two triangles $\triangle ABC$ and $\triangle DEF$ such that $AB = DE$, $AC = DF$, and $\angle BAC = \angle EDF$. 3. **Goal:** Show that $\triangle ABC \cong \triangle DEF$. 4. **Proof:** - Place $\triangle ABC$ on a coordinate plane with $A$ at the origin $(0,0)$. - Let $AB$ lie along the positive x-axis, so $B$ is at $(AB,0)$. - Since $AC = DF$ and $\angle BAC = \angle EDF$, point $C$ lies at coordinates $(AC \cos \theta, AC \sin \theta)$ where $\theta = \angle BAC$. - Similarly, place $\triangle DEF$ with $D$ at the origin and $DE$ along the x-axis. - Then $E$ is at $(DE,0)$ and $F$ at $(DF \cos \theta, DF \sin \theta)$. - Since $AB = DE$, $AC = DF$, and $\angle BAC = \angle EDF$, points $B$ and $E$ coincide, and points $C$ and $F$ coincide. - Therefore, all corresponding sides and angles match, proving $\triangle ABC \cong \triangle DEF$ by SAS. 5. **Conclusion:** The SAS theorem is proven by placing triangles in a coordinate system and showing that the given equal sides and included angle force the triangles to coincide exactly.