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:

Step1: Analyze given triangle info

We know $AC \cong QR$, $\angle C \cong \angle Q$ from the diagram.

Step2: Evaluate Option 1 (∠A≅∠T, ASA)

ASA needs 2 angles + included side. Here, $\angle A \cong \angle T$, $\angle C \cong \angle Q$, $AC \cong QR$ (included side for angles). This fits ASA.

Step3: Evaluate Option 2 (∠B≅∠P, AAS)

AAS needs 2 angles + non-included side. We have $\angle B \cong \angle P$, $\angle C \cong \angle Q$, $AC \cong QR$. This fits AAS.

Step4: Evaluate Option 3 (All acute angles)

Acute angles alone don't prove congruence; triangles can be similar but not congruent.

Step5: Evaluate Option 4 (∠C,∠Q are right angles)

Only knowing they are right angles (we already know $\angle C \cong \angle Q$) plus one side is not enough for congruence.

Step6: Evaluate Option 5 (BC≅PQ, ASA)

$BC \cong PQ$ would not form an included side for the known angles, so this does not fit ASA.

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.