Question

what additional information could be used to prove that the triangles are congruent using aas or asa? select three options. ∠b ≅ ∠p and \\(\overline{bc} \cong \overline{pq}\\); ∠a ≅ ∠t and \\(ac = tq = 3.2\text{cm}\\); ∠a ≅ ∠t and ∠b ≅ ∠p; ∠a ≅ ∠t and \\(\overline{bc} \cong \overline{pq}\\); \\(ac = tq = 3.2\text{ cm}\\) and \\(cb = qp = 2.2\text{ cm}\\)

Explanation:

1. For $\angle B \cong \angle P$ and $\overline{BC} \cong \overline{PQ}$: We can use AAS or ASA. If we consider the angles and the included or non - included side, with $\angle C$ and $\angle Q$ (from the right angles, assuming the right - angled triangles) and the given angle and side, this can help prove congruence.
2. For $\angle A \cong \angle T$ and $AC = TQ=3.2$ cm: If $\angle A\cong\angle T$, and we know $AC = TQ$, and we can consider the right angles (if the triangles are right - angled) or other angles, this can be used in ASA or AAS.
3. For $\angle A \cong \angle T$ and $\overline{BC} \cong \overline{PQ}$: With $\angle A\cong\angle T$, $\overline{BC}\cong\overline{PQ}$ and the right angles (or other common angles), we can apply AAS or ASA to prove congruence.

The option $\angle A \cong \angle T$ and $\angle B \cong \angle P$ only gives us two angles, which is AA (Angle - Angle) and we need a side to prove congruence using AAS or ASA. The option $AC = TQ = 3.2$ cm and $CB=QP = 2.2$ cm is SSS (Side - Side - Side) or SAS (Side - Angle - Side) related, not AAS or ASA.

Answer:

• $\boldsymbol{\angle B \cong \angle P}$ and $\boldsymbol{\overline{BC} \cong \overline{PQ}}$
• $\boldsymbol{\angle A \cong \angle T}$ and $\boldsymbol{AC = TQ = 3.2}$ cm
• $\boldsymbol{\angle A \cong \angle T}$ and $\boldsymbol{\overline{BC} \cong \overline{PQ}}$