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 ASA (Angle - Side - Angle)

ASA requires two angles and the included side. If \(\angle A\cong\angle T\), we don't have the included side situation.

Step2: Analyze AAS (Angle - Angle - Side)

We know that \(AC = QT\) (given as equal from the markings). If \(\angle B\cong\angle P\), we have two angles (\(\angle B\cong\angle P\) and the given non - included angles) and a non - included side (\(AC = QT\)), which satisfies AAS.

Step3: Analyze the "all angles acute"

Just knowing all angles are acute is not a congruence criterion. Congruence criteria are SSS, SAS, ASA, AAS, HL (for right - angled triangles).

Step4: Analyze right - angle case

If \(\angle C\) and \(\angle Q\) are right angles, we still don't have enough information (like hypotenuse - leg or other side - angle combinations) to prove congruence.

Step5: Analyze \(BC\cong PQ\)

If \(BC\cong PQ\), we have \(AC = QT\) (given), \(\angle C=\angle Q\) (from the markings of the angles in the triangles). So, by ASA (\(AC = QT\), \(\angle C=\angle Q\), \(BC\cong PQ\))

Answer:

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