Question

1. hl

a) (angle prqcongangle wqr)
b) (overline{rq}congoverline{qr})
c) (angle rqpcongangle qrw)
d) (overline{qp}congoverline{rw})

Explanation:

Step1: Analyze the Hypotenuse - Leg (HL) Congruence Theorem

The HL theorem states that if the hypotenuse and a leg of one right - triangle are congruent to the hypotenuse and a leg of another right - triangle, then the two right - triangles are congruent. In right - triangles \( \triangle PRQ\) and \( \triangle WQR\), for HL to be applicable, we need to check the congruence of corresponding parts.

• For option A: \( \angle PRQ\) and \( \angle WQR\) are right - angles (already given as right - angles in the figure, not part of the HL criteria which is about sides).
• For option B: \( \overline{RQ}\) and \( \overline{QR}\) are the same segment. In the context of triangle congruence (HL for right - triangles \(\triangle PRQ\) and \(\triangle WQR\)), \(RQ\) is a leg of \(\triangle PRQ\) and \(QR\) is a leg of \(\triangle WQR\). Since \(RQ = QR\) (reflexive property of equality, and in terms of congruence \(\overline{RQ}\cong\overline{QR}\)), and if we assume the hypotenuses are congruent (implied in the HL problem setup), this is a valid part of the HL criteria.
• For option C: \( \angle RQP\) and \( \angle QRW\) are not relevant to the HL (side - side) criteria for right - triangle congruence.
• For option D: \( \overline{QP}\) and \( \overline{RW}\) are not hypotenuses or legs in the correct correspondence for the HL theorem.

Answer:

B. \( \overline{RQ}\cong\overline{QR}\)