Question

select the correct answer.
given: \\(\overline{pr}\\) bisects \\(\angle qps\\), \\(pq = 12\\) units, and \\(ps = 12\\) units
prove: \\(\overline{qr} \cong \overline{sr}\\)
image of a geometric figure with points p, q, r, s
how could you show \\(\overline{qr} \cong \overline{sr}\\)?
\\(\bigcirc\\) a. use sas to show triangle prq is congruent to triangle prs
\\(\bigcirc\\) b. use sas to show triangle prq is congruent to triangle srp
\\(\bigcirc\\) c. use aas to show triangle prq is congruent to triangle prs
\\(\bigcirc\\) d. use asa to show triangle prq is congruent to triangle prs

Explanation:

1. Analyze given information:

• $\overline{PR}$ bisects $\angle QPS$, so $\angle QPR=\angle SPR$.
• $PQ = PS = 12$ units.
• $\overline{PR}$ is common to both $\triangle PRQ$ and $\triangle PRS$.

1. Check congruence criteria:

• For SAS (Side - Angle - Side), we need two sides and the included angle.
• In $\triangle PRQ$ and $\triangle PRS$:
• $PQ = PS$ (given, side).
• $\angle QPR=\angle SPR$ (angle bisector, included angle).
• $PR = PR$ (common side).
• So we can use SAS to prove $\triangle PRQ\cong\triangle PRS$.
• Option B is incorrect because the triangles compared ($\triangle PRQ$ and $\triangle SRP$) do not have corresponding sides/angles correctly. Option C (AAS) and D (ASA) are incorrect as we have two sides and the included angle, not AAS or ASA.

Answer:

A. Use SAS to show triangle PRQ is congruent to triangle PRS