Question

use the given information to prove that ( overline{ps} cong overline{rs} ).
given: ( overline{qs} perp overline{pr} )
( overline{qs} ) bisects ( angle pqr )
prove: ( overline{ps} cong overline{rs} )

Explanation:

Step1: Define right angles

Since \( \overline{QS}\perp\overline{PR}\), then \( \angle PSQ = \angle RSQ=90^{\circ}\) (Definition of perpendicular lines).

Step2: Use angle - bisector property

Because \( \overline{QS}\) bisects \( \angle PQR\), so \( \angle PQS=\angle RQS\) (Definition of angle - bisector).

Step3: Use common side

\( \overline{QS}\cong\overline{QS}\) (Reflexive property of congruence).

Step4: Prove triangles congruent

By the Angle - Angle - Side (AAS) congruence criterion, \( \triangle PQS\cong\triangle RQS\) (\( \angle PQS=\angle RQS\), \( \angle PSQ = \angle RSQ\), \( \overline{QS}\cong\overline{QS}\)).

Step5: Conclude segment congruence

Since \( \triangle PQS\cong\triangle RQS\), then \( \overline{PS}\cong\overline{RS}\) (Corresponding parts of congruent triangles are congruent).

Answer:

\(\overline{PS}\cong\overline{RS}\) is proved as above.