Right Triangle Congruence Fb2077

1. **Problem Statement:** Determine if the reason for congruence of two right triangles is always Hypotenuse-Leg (HL). 2. **Important Concept:** For right triangles, the Hypotenuse-Leg (HL) theorem states that if the hypotenuse and one leg of one right triangle are congruent to the hypotenuse and one leg of another right triangle, then the triangles are congruent. 3. **Key Point:** While HL is a valid congruence criterion for right triangles, it is not the only one. Right triangles can also be congruent by other criteria such as Side-Angle-Side (SAS) or Angle-Side-Angle (ASA), provided the right angle is included. 4. **Example 1 (HL):** Given two right triangles $\triangle ABC$ and $\triangle DEF$ with right angles at $C$ and $F$ respectively. If $AB = DE$ (hypotenuses) and $AC = DF$ (one leg), then by HL theorem, $\triangle ABC \cong \triangle DEF$. 5. **Example 2 (SAS):** Given two right triangles $\triangle XYZ$ and $\triangle PQR$ with right angles at $Z$ and $R$ respectively. If $XZ = PQ$ (leg), $\angle X = \angle P = 90^\circ$, and $YZ = QR$ (leg), then by SAS theorem, $\triangle XYZ \cong \triangle PQR$. 6. **Conclusion:** The reason for congruence of two right triangles is not always HL. Other congruence criteria like SAS and ASA can also apply if the right angle is included. **Final answer:** No, the reason is not always Hypotenuse-Leg (HL).