Question
Triangles A B C and T P Q are shown. Angles B C A and P Q T are congruent.
What additional information could be used to prove that the triangles are congruent using AAS or ASA? Select three options.
AngleB ≅ AngleP and BC ≅ PQ
AngleA ≅ AngleT and AC = TQ = 3.2cm
AngleA ≅ AngleT and AngleB ≅ AngleP
AngleA ≅ AngleT and BC ≅ PQ
AC = TQ = 3.2 cm and CB = QP = 2.2 cm
What additional information could be used to prove that the triangles are congruent using AAS or ASA? Select three options.
AngleB ≅ AngleP and BC ≅ PQ
AngleA ≅ AngleT and AC = TQ = 3.2cm
AngleA ≅ AngleT and AngleB ≅ AngleP
AngleA ≅ AngleT and BC ≅ PQ
AC = TQ = 3.2 cm and CB = QP = 2.2 cm
Answers
Answered by
GPT-4o mini
To prove that triangles ABC and TPQ are congruent using the Angle-Angle-Side (AAS) or Angle-Side-Angle (ASA) criteria, we need to show that we have two angles and the included side (in the case of ASA) or an angle and a non-included side (in the case of AAS).
Given that angles \( \angle BCA \) and \( \angle PQT \) are congruent, we need to add two more pieces of information from the options provided.
The best options to use are:
1. **Angle A ≅ Angle T and Angle B ≅ Angle P**: This option provides the necessary angles to establish congruence using the AAS theorem, with the sides being the respective sides opposite those angles.
2. **Angle A ≅ Angle T and BC ≅ PQ**: This option allows us to use AAS as well. We have angle A and angle T as congruent and the side BC equal to PQ.
3. **Angle A ≅ Angle T and AC = TQ = 3.2cm**: This option also offers two angles and a side that is not included (AAS) that allows us to establish the congruence of the triangles.
In summary, the three options that would prove the triangles are congruent are:
1. **Angle A ≅ Angle T and Angle B ≅ Angle P**
2. **Angle A ≅ Angle T and BC ≅ PQ**
3. **Angle A ≅ Angle T and AC = TQ = 3.2cm**
Given that angles \( \angle BCA \) and \( \angle PQT \) are congruent, we need to add two more pieces of information from the options provided.
The best options to use are:
1. **Angle A ≅ Angle T and Angle B ≅ Angle P**: This option provides the necessary angles to establish congruence using the AAS theorem, with the sides being the respective sides opposite those angles.
2. **Angle A ≅ Angle T and BC ≅ PQ**: This option allows us to use AAS as well. We have angle A and angle T as congruent and the side BC equal to PQ.
3. **Angle A ≅ Angle T and AC = TQ = 3.2cm**: This option also offers two angles and a side that is not included (AAS) that allows us to establish the congruence of the triangles.
In summary, the three options that would prove the triangles are congruent are:
1. **Angle A ≅ Angle T and Angle B ≅ Angle P**
2. **Angle A ≅ Angle T and BC ≅ PQ**
3. **Angle A ≅ Angle T and AC = TQ = 3.2cm**