A triangle is made up of vertices upper A, upper B, and upper C. Another triangle is made up of vertices upper P, upper Q, and upper R. Angles upper C, upper A, upper Q, and upper R are marked by arcs.

Question

A triangle is made up of vertices upper A, upper B, and upper C. Another triangle is made up of vertices upper P, upper Q, and upper R. Angles upper C, upper A, upper Q, and upper R are marked by arcs.
Based on the given information in the diagram, what additional information is needed to prove  △ABC≅△RPQ
  by the AAS Congruence Theorem? Enter the number of the correct option.

Option #1:  AC¯¯¯¯¯¯¯¯≅RQ¯¯¯¯¯¯¯¯

Option #2:  ∠B≅∠P

Option #3:  BC¯¯¯¯¯¯¯¯≅PQ¯¯¯¯¯¯¯¯

(1 point)
The additional information needed is Option #
.

GPT-4o mini · Answer

To prove triangles $ \triangle ABC $ and $ \triangle RPQ $ are congruent using the AAS (Angle-Angle-Side) Congruence Theorem, you need two angles and the length of one side in one triangle to be congruent to the corresponding angles and side in the other triangle. From the description, angles $ \angle C $, $ \angle A $, $ \angle Q $, and $ \angle R $ are marked, which indicates we have two pairs of congruent angles (assuming the pairs were indicated as congruent in arcs). To apply the AAS theorem, we need to know the length of one side between those angles. The options provided are: - Option #1: $ AC \cong RQ $ - Option #2: $ \angle B \cong \angle P $ - Option #3: $ BC \cong PQ $ For AAS to hold, we actually would require either Option #2 (providing that we have confirmed $ \angle B \cong \angle P $) along with the other provided angles, and a side joining the two angles, like $ AC $ or $ BC $. The most direct option needed to establish AAS from the options presented is **Option #2: $ \angle B \cong \angle P $**. Thus, the answer is: The additional information needed is Option #2.