Question

select the correct answer. given: $overline{pr}$ bisects $angle qps$, $pq = 12$ units, and $ps = 12$ units. prove: $overline{qr}congoverline{sr}$. how could you show $overline{qr}congoverline{sr}$? a. use sas to show triangle $prq$ is congruent to triangle $srp$. b. use asa to show triangle $prq$ is congruent to triangle $prs$. c. use aas to show triangle $prq$ is congruent to triangle $prs$. d. use sas to show triangle $prq$ is congruent to triangle $prs$.

Explanation:

Step1: Recall angle - bisector property

Since $\overline{PR}$ bisects $\angle QPS$, then $\angle QPR=\angle SPR$.

Step2: Identify given side - lengths

We are given that $PQ = 12$ units and $PS=12$ units, so $PQ = PS$.

Step3: Identify common side

$\overline{PR}$ is common to both $\triangle PRQ$ and $\triangle PRS$, i.e., $PR = PR$.

Step4: Apply congruence criterion

By the Side - Angle - Side (SAS) congruence criterion (two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle), $\triangle PRQ\cong\triangle PRS$.

Step5: Use CPCTC

Corresponding Parts of Congruent Triangles are Congruent (CPCTC). Since $\triangle PRQ\cong\triangle PRS$, then $\overline{QR}\cong\overline{SR}$. The correct way to show the congruence of the triangles to get $\overline{QR}\cong\overline{SR}$ is to use SAS to show $\triangle PRQ$ is congruent to $\triangle PRS$.

Answer:

D. Use SAS to show triangle $PRQ$ is congruent to triangle $PRS$