Question

which statements are true about additional information for proving that the triangles are congruent? select two options.
if ∠a ≅ ∠t, then the triangles would be congruent by asa.
if ∠b ≅ ∠p, then the triangles would be congruent by aas.
if all the angles are acute, then the triangles would be congruent.
if ∠c and ∠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 the first option: We know \( AC \cong QT \) (marked with a tick) and \( \angle C \cong \angle Q \) (marked angles). If \( \angle A \cong \angle T \), then by ASA (Angle - Side - Angle: two angles and the included side), \( \triangle ABC \cong \triangle TQP \). So this statement is true.
2. Analyze the second option: Given \( AC \cong QT \), \( \angle C \cong \angle Q \). If \( \angle B \cong \angle P \), then by AAS (Angle - Angle - Side: two angles and a non - included side), \( \triangle ABC \cong \triangle TPQ \). So this statement is true.
3. Analyze the third option: Just knowing all angles are acute does not guarantee congruence. Triangles can have the same angle measures (be similar) but different side lengths. So this statement is false.
4. Analyze the fourth option: If \( \angle C \) and \( \angle Q \) are right angles, we know \( AC \cong QT \), but we don't know if another side or angle is congruent. We can't conclude congruence just from right angles and one pair of congruent sides. So this statement is false.
5. Analyze the fifth option: If \( \overline{BC} \cong \overline{PQ} \), with \( \angle C \cong \angle Q \) and \( AC \cong QT \), the congruence criterion would be SAS (Side - Angle - Side), not ASA. So this statement is false.

Answer:

• 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.