1. **Problem Statement:** Prove that if two right triangles have one leg and the hypotenuse respectively equal, then the triangles are congruent. This is known as the Right-Angle-Side-Side (RASS) theorem. 2. **Given:** Two right triangles $\triangle ABC$ and $\triangle DEF$ with right angles at $C$ and $F$ respectively. - One leg equal: $AC = DF$ - Hypotenuse equal: $AB = DE$ 3. **To Prove:** $\triangle ABC \cong \triangle DEF$ 4. **Key Idea:** In right triangles, if the hypotenuse and one leg are equal, the triangles are congruent by the Hypotenuse-Leg (HL) theorem, which is a special case of the Side-Angle-Side (SAS) congruence. 5. **Proof Steps:** - Both triangles have a right angle, so $\angle C = \angle F = 90^\circ$. - Given $AC = DF$ (one leg equal) and $AB = DE$ (hypotenuse equal). - By the Pythagorean theorem, the other leg lengths are determined uniquely: $$BC = \sqrt{AB^2 - AC^2}$$ $$EF = \sqrt{DE^2 - DF^2}$$ - Since $AB = DE$ and $AC = DF$, it follows that: $$BC = EF$$ - Therefore, all corresponding sides are equal: $$AB = DE, \quad AC = DF, \quad BC = EF$$ - By the Side-Side-Side (SSS) congruence postulate, $\triangle ABC \cong \triangle DEF$. 6. **Conclusion:** The two right triangles are congruent if one leg and the hypotenuse are equal, proving the RASS theorem.