Question

in the given triangles, \\( \overline{pr} \cong \overline{ac} \\) and \\( \angle p \cong \angle a \\). which additional fact is needed in order to use the asa criterion to prove that the two triangles are congruent? a. \\( \angle r \cong \angle c \\) b. \\( \overline{pq} \cong \overline{ab} \\) c. \\( \overline{rq} \cong \overline{bc} \\) d. \\( \angle q \cong \angle b \\)

Explanation:

Step1: Recall ASA Criterion

ASA (Angle - Side - Angle) criterion for triangle congruence states that if two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent.

We know that \(PR\cong AC\) (a side) and \(\angle P\cong\angle A\) (an angle). For ASA, we need another pair of angles such that the side \(PR\) (or \(AC\)) is included between the two angles.

Step2: Analyze Each Option

• Option A: \(\angle R\cong\angle C\). Let's see the correspondence. In \(\triangle PQR\) and \(\triangle ABC\), we have \(\angle P\cong\angle A\), \(PR\cong AC\), and if \(\angle R\cong\angle C\), then we have two angles (\(\angle P\) and \(\angle R\)) and the included side \(PR\) in \(\triangle PQR\) congruent to two angles (\(\angle A\) and \(\angle C\)) and the included side \(AC\) in \(\triangle ABC\). This satisfies ASA.
• Option B: \(PQ\cong AB\) is a side - side - angle or side - angle - side related, not ASA.
• Option C: \(RQ\cong BC\) is also a side - related condition, not related to ASA.
• Option D: \(\angle Q\cong\angle B\) would be for AAS (Angle - Angle - Side) if combined with other conditions, not ASA.

Answer:

A. \(\angle R\cong\angle C\)