Question

which statements are true about additional information for proving that the triangles are congruent? select two options.
if (angle acongangle t), then the triangles would be congruent by asa.
if (angle bcongangle p), then the triangles would be congruent by aas.
if all the angles are acute, then the triangles would be congruent.
if (angle c) and (angle q) are right angles, then triangles would be congruent.
if (overline{bc}congoverline{pq}), then the triangles would be congruent by asa.

Explanation:

Step1: Recall congruence criteria

ASA (Angle - Side - Angle): Two angles and the included side are congruent. AAS (Angle - Angle - Side): Two angles and a non - included side are congruent.

Step2: Analyze each option

• For “If \(∠A\cong∠T\), then the triangles would be congruent by ASA”:

We don't know if the sides between the angles are congruent. So this is false.

• For “If \(∠B\cong∠P\)”:

We already have \(∠C\cong∠Q\) (given by the arc markings in the figure) and assume a side (by the structure of AAS, if we consider the non - included side). So by AAS (since we have two angles and a non - included side), the triangles can be congruent.

• For “If all the angles are acute, then the triangles would be congruent”:

Just having all angles acute doesn't guarantee congruence. Similarity (by AA) is possible but not congruence. So this is false.

• For “If \(∠C\) and \(∠Q\) are right angles”:

Just knowing they are right angles (adding to the existing angle information) doesn't give enough side or other angle information for congruence. So this is false.

• For “If \(\overline{BC}\cong\overline{PQ}\)”:

We have \(∠C\cong∠Q\) (given by arc markings) and assume another angle (say \(∠B\) and \(∠P\) or \(∠A\) and \(∠T\) based on the figure's angle - side - angle structure). By ASA (if the side is between two angles), the triangles can be congruent.

Answer:

If \(∠B\cong∠P\), then the triangles would be congruent by AAS; If \(\overline{BC}\cong\overline{PQ}\), then the triangles would be congruent by ASA.