Which of the following conditions must be true for two right triangles to be congruent by the Hypotenuse-Leg ($\mathrm{HL}$) theorem?

• A. The triangles must each have a right angle, their hypotenuses must be $\cong$ (congruent), and one pair of corresponding legs must be $\cong$.
• B. The triangles must have two pairs of $\cong$ angles and one pair of $\cong$ sides.
• C. The triangles must have two pairs of $\cong$ sides and the included angle $\cong$.
• D. The triangles must have all three sides $\cong$.

Answer: (A)

Verified step by step guidance

1. Understand that the Hypotenuse-Leg (HL) theorem applies specifically to right triangles, so the first condition is that both triangles must have a right angle.
2. Recognize that in right triangles, the hypotenuse is the side opposite the right angle and is the longest side. For HL congruence, the hypotenuses of both triangles must be congruent (equal in length).
3. Identify that besides the hypotenuse, one corresponding leg (one of the two sides forming the right angle) must also be congruent between the two triangles.
4. Recall that the HL theorem states: If two right triangles have congruent hypotenuses and one pair of congruent legs, then the triangles are congruent.
5. Compare this with other congruence criteria like ASA, SAS, and SSS, which require different combinations of angles and sides, confirming that HL is unique to right triangles with the conditions above.