Question

given: \\( \angle prs \\) and \\( \angle rpq \\) are right angles
\\( \overline { sp } \cong \overline { qr } \\)
prove: \\( \triangle prs \cong \triangle rpq \\)

Explanation:

Step1: State the right - angle property

Since \(\angle PRS\) and \(\angle RPQ\) are right angles, \(\angle PRS = 90^{\circ}\) and \(\angle RPQ=90^{\circ}\) (Given).

Step2: Use the given congruent segments

We are given that \(\overline{SP}\cong\overline{QR}\).

Step3: Use the reflexive property

For the side \(\overline{PR}\), by the reflexive property of congruence, \(\overline{PR}\cong\overline{PR}\).

Step4: Apply the Hypotenuse - Leg (HL) Congruence Theorem

In right - triangles \(\triangle PRS\) and \(\triangle RPQ\), the hypotenuse - leg (HL) congruence 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. Here, in \(\triangle PRS\) and \(\triangle RPQ\), \(\overline{SP}\) (hypotenuse of \(\triangle PRS\)) \(\cong\overline{QR}\) (hypotenuse of \(\triangle RPQ\)) and \(\overline{PR}\) (common leg) \(\cong\overline{PR}\). So, \(\triangle PRS\cong\triangle RPQ\) by the HL congruence theorem.

Answer:

The proof is completed as shown in the steps above, with the final conclusion that \(\triangle PRS\cong\triangle RPQ\) by the Hypotenuse - Leg (HL) Congruence Theorem.