To determine which statements are true about additional information for proving that triangles ABC and TPQ are congruent, we analyze each option given the information that sides AC and TQ are congruent, and angles BCA and PQT are congruent.
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If Angle A ≅ Angle T, then the triangles would be congruent by ASA.
- This statement is true. If you have two angles and the included side (AC ≅ TQ), then the triangles can be proven congruent by the Angle-Side-Angle (ASA) postulate.
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If Angle B ≅ Angle P, then the triangles would be congruent by AAS.
- This statement is true. If Angle B is congruent to Angle P and you also have one side (AC ≅ TQ) and another angle (∠BCA ≅ ∠PQT) that are congruent, then by the Angle-Angle-Side (AAS) theorem, the triangles would be congruent.
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If all the angles are acute, then the triangles would be congruent.
- This statement is false. The fact that all angles are acute does not provide enough information to establish congruence without knowing the relationships of the sides.
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If Angle C and Angle Q are right angles, then triangles would be congruent.
- This statement is false because, while both triangles would be right triangles, it does not automatically guarantee that they are congruent since we need to ensure that the other corresponding sides or angles are also congruent.
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If BC ≅ PQ, then the triangles would be congruent by ASA.
- This statement is false. To use ASA, we would need the congruence of an angle between the two sides. While having BC ≅ PQ along with the previous angle congruence would provide information about triangle congruence, it does not lead directly to congruence by ASA alone.
Therefore, the correct statements about additional information for proving that the triangles are congruent are:
- If Angle A ≅ Angle T, then the triangles would be congruent by ASA.
- If Angle B ≅ Angle P, then the triangles would be congruent by AAS.
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