Question

given: ∠prs and ∠rpq are right angles
sp ≅ qr
prove: △prs ≅ △rpq

Explanation:

Step1: State the given information

• $\angle PRS$ and $\angle RPQ$ are right angles. So, $\angle PRS=\angle RPQ = 90^{\circ}$.
• $\overline{SP}\cong\overline{QR}$ (Given).
• $\overline{PR}\cong\overline{PR}$ (Reflexive property of congruence: For any segment $AB$, $AB\cong AB$).

Step2: Apply the Hypotenuse - Leg (HL) congruence criterion

In right - triangles $\triangle PRS$ and $\triangle RPQ$, for a right - triangle, the HL (Hypotenuse - Leg) 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 $\triangle RPQ$, $\overline{SP}$ and $\overline{QR}$ are legs, $\overline{PR}$ is the common hypotenuse.

Answer:

Statements	Reasons
$\overline{SP}\cong\overline{QR}$	Given
$\overline{PR}\cong\overline{PR}$	Reflexive property of congruence
$\triangle PRS\cong\triangle RPQ$	Hypotenuse - Leg (HL) congruence criterion