Question

if $overline{ps} cong overline{rs}$ and $mangle pqs = 48^circ$, what is $mangle rqs$?

$mangle rqs = \square^circ$

Explanation:

Step1: Identify congruent triangles

We have \( \overline{PS} \cong \overline{RS} \), \( \angle QPS = \angle QRS = 90^\circ \) (right angles), and \( \overline{QS} \) is common to both triangles \( \triangle QPS \) and \( \triangle QRS \). By the Hypotenuse - Leg (HL) congruence criterion, \( \triangle QPS \cong \triangle QRS \).

Step2: Use angle bisector property

Since \( \triangle QPS \cong \triangle QRS \), the corresponding angles \( \angle PQS \) and \( \angle RQS \) are equal. We know that \( m\angle PQS = 48^\circ \), so \( m\angle RQS=m\angle PQS = 48^\circ \).

Answer:

\( 48 \)