Question

1. $overline{qs}$ is perpendicular to $overline{rp}$, and $overline{qs}$ is an angle bisector of $angle pqr$.

can you prove $\triangle pqs cong \triangle rqs$? justify your answer.

$\bigcirc$ no, by sas criteria: $overline{pq} cong overline{qr}$, $angle pqs cong angle rqs$, and shared side $overline{qr}$ is not congruent to itself.

$\bigcirc$ no, by sas criteria: $overline{pq} cong overline{qr}$, $angle pqs cong angle rqs$, and shared side $overline{qs}$ is not congruent to itself.

$\bigcirc$ yes, by sas criteria: $overline{pq} cong overline{qs}$, $angle pqs cong angle rqs$, and shared side $overline{qr}$ is congruent to itself.

$\bigcirc$ yes, by sas criteria: $overline{pq} cong overline{qr}$, $angle pqs cong angle rqs$, and shared side $overline{qs}$ is congruent to itself.

Explanation:

Step1: Identify given congruent sides

$\overline{PQ} \cong \overline{QR}$ (marked on diagram, from angle bisector + perpendicular implies isosceles triangle)

Step2: Identify given congruent angles

$\angle PQS \cong \angle RQS$ ($\overline{QS}$ is angle bisector)

Step3: Identify shared congruent side

$\overline{QS} \cong \overline{QS}$ (reflexive property, shared side)

Step4: Apply SAS congruence rule

Two sides and included angle of $\triangle PQS$ match $\triangle RQS$, so $\triangle PQS \cong \triangle RQS$ by SAS.

Answer:

Yes, by SAS criteria: $\overline{PQ} \cong \overline{QR}, \angle PQS \cong \angle RQS$, and shared side $\overline{QS}$ is congruent to itself.