Question

if $\angle pqs \cong \angle rqs$ and $ps = 79$, what is $rs$?
$rs = \square$

Explanation:

Step1: Identify the theorem

We use the Angle - Bisector Theorem for right triangles (or the property of angle - bisector and perpendiculars). Since \(\angle PQS\cong\angle RQS\), \(QS\) is the angle bisector of \(\angle PQR\). Also, \(SP\perp QP\) and \(SR\perp QR\) (because of the right angles at \(P\) and \(R\)). By the property of angle - bisectors, the distance from a point on the angle bisector to the two sides of the angle is equal. So, \(RS = PS\).

Step2: Substitute the value of \(PS\)

We know that \(PS = 79\). From the property we derived in Step 1, \(RS=PS\).

Answer:

\(79\)