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 bc ≅ pq, then the triangles would be congruent by asa.

Explanation:

Step1: Recall congruence criteria

• ASA (Angle - Side - Angle): Two angles and the included side of one triangle are equal to two angles and the included side of another triangle.
• AAS (Angle - Angle - Side): Two angles and a non - included side of one triangle are equal to two angles and the corresponding non - included side of another triangle.

Step2: Analyze each option

• Option 1:

If \(\angle A\cong\angle T\), we have \(\angle C\cong\angle Q\) (given by the arc in the figure) and \(AC = QI\) (marked as equal). But for ASA, the side should be included between the two angles. Here, the side is not included, so this is incorrect.

• Option 2:

If \(\angle B\cong\angle P\), we know \(\angle C\cong\angle Q\) (given by the arc) and \(AC = QI\) (marked as equal). By AAS (two angles and a non - included side), the triangles would be congruent.

• Option 3:

Just because all angles are acute does not guarantee triangle congruence. Congruence requires a specific relationship between sides and angles (like SSS, SAS, ASA, AAS, HL for right - angled triangles), not just the nature of angles (acute, obtuse, right). So this is incorrect.

• Option 4:

If \(\angle C\) and \(\angle Q\) are right angles, we still need information about the sides. Just having right angles (one pair of equal angles) is not sufficient for congruence. So this is incorrect.

• Option 5:

If \(BC\cong PQ\), we know \(\angle C\cong\angle Q\) (given by the arc) and \(AC = QI\) (marked as equal). By ASA (angle - side - angle, where \(\angle C\cong\angle Q\), \(BC\cong PQ\), and \(\angle B\) and \(\angle P\) can be related as we have two angles and the included side conceptually if we consider the side between the two angles.

Answer:

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