Question

which statements are true about additional information for proving that the triangles are congruent? select two options.

• if $\angle a \cong \angle t$, then the triangles would be congruent by asa.
• if $\angle b \cong \angle p$, 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 $\overline{bc} \cong \overline{pq}$, then the triangles would be congruent by asa.

Explanation:

1. Analyze each option:

• Option 1: If \( \angle A \cong \angle T \), we have a side (marked congruent), an angle (the marked angle at \( C \) and \( Q \)), and another angle (\( \angle A \) and \( \angle T \)), which is ASA (Angle - Side - Angle) congruence criterion. So this statement is true.
• Option 2: If \( \angle B \cong \angle P \), we have a side (marked congruent), the marked angle (at \( C \) and \( Q \)), and another angle (\( \angle B \) and \( \angle P \)), which is AAS (Angle - Angle - Side) congruence criterion. So this statement is true.
• Option 3: Just knowing all angles are acute does not guarantee congruence. Triangles with all acute angles can be similar but not congruent (different side lengths). So this statement is false.
• Option 4: If \( \angle C \) and \( \angle Q \) are right angles, we have a right angle, a marked side, but we need more information (like another angle or side) to prove congruence. Just right angles and one marked side are not enough. So this statement is false.
• Option 5: If \( \overline{BC} \cong \overline{PQ} \), we have a side, the marked angle, but the ASA criterion requires the side to be between the two angles. Here, the side \( BC \) and \( PQ \) are not between the correct angles for ASA. So this statement is false.

Answer:

A. If \( \angle A \cong \angle T \), then the triangles would be congruent by ASA.
B. If \( \angle B \cong \angle P \), then the triangles would be congruent by AAS.