Question

what additional information could be used to prove that the triangles are congruent using aas or asa? choose three correct answers.
∠a ≅ ∠t and
bc ≅ pq
∠a ≅ ∠t and
∠b ≅ ∠p
∠b ≅ ∠p and
bc ≅ pq
∠a ≅ ∠t and
ac = tq = 3.2 cm
ac = tq = 3.2 cm and
cb = qp = 2.2 cm

Explanation:

Step1: Recall AAS/ASA Congruence

AAS (Angle - Angle - Side) and ASA (Angle - Side - Angle) are triangle congruence criteria. ASA requires two angles and the included side to be congruent, AAS requires two angles and a non - included side to be congruent. From the diagram, we know that $\angle C\cong\angle Q$.

Step2: Analyze Each Option

• Option 1 ($\angle A\cong\angle T$ and $\overline{BC}\cong\overline{PQ}$): We have $\angle C\cong\angle Q$, $\angle A\cong\angle T$ and side $BC\cong PQ$ (non - included side between the two angles). This satisfies AAS.
• Option 2 ($\angle A\cong\angle T$ and $\angle B\cong\angle P$): Having three angles congruent (AAA) does not prove triangle congruence, it only proves similarity. So this is incorrect.
• Option 3 ($\angle B\cong\angle P$ and $\overline{BC}\cong\overline{PQ}$): We have $\angle C\cong\angle Q$, $\angle B\cong\angle P$ and side $BC\cong PQ$ (included side between $\angle B$ and $\angle C$ for $\triangle ABC$ and between $\angle P$ and $\angle Q$ for $\triangle TPQ$). This satisfies ASA.
• Option 4 ($\angle A\cong\angle T$ and $AC = TQ=3.2$ cm): We have $\angle C\cong\angle Q$, $\angle A\cong\angle T$ and side $AC\cong TQ$ (non - included side between the two angles). This satisfies AAS.
• Option 5 ($AC = TQ = 3.2$ cm and $CB=QP = 2.2$ cm): This is SSS (Side - Side - Side) or SAS (if we consider the included angle), but not AAS or ASA. So this is incorrect.

Answer:

1. $\angle A\cong\angle T$ and $\overline{BC}\cong\overline{PQ}$
2. $\angle B\cong\angle P$ and $\overline{BC}\cong\overline{PQ}$
3. $\angle A\cong\angle T$ and $AC = TQ = 3.2$ cm