Question

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

Explanation:

Step1: Recall AAS/ASA rules

AAS: 2 angles + non-included side ≅; ASA: 2 angles + included side ≅.

Step2: Analyze Option 1

$\angle B \cong \angle P$ and $\overline{BC} \cong \overline{PQ}$: $\angle B$ corresponds to $\angle P$, $\overline{BC}$ (non-included side for $\angle A, \angle B$) corresponds to $\overline{PQ}$ (non-included side for $\angle T, \angle P$). This fits AAS.

Step3: Analyze Option 2

$\angle A \cong \angle T$ and $AC = TQ = 3.2$cm: $\angle A$ corresponds to $\angle T$, $\overline{AC}$ (non-included side for $\angle A, \angle B$) corresponds to $\overline{TQ}$ (non-included side for $\angle T, \angle P$). This fits AAS.

Step4: Analyze Option 3

$\angle A \cong \angle T$ and $\angle B \cong \angle P$: Only 2 angles congruent, no side given. Cannot prove congruence with AAS/ASA.

Step5: Analyze Option 4

$\angle A \cong \angle T$ and $\overline{BC} \cong \overline{PQ}$: $\angle A$ corresponds to $\angle T$, $\overline{BC}$ (opposite $\angle A$) corresponds to $\overline{PQ}$ (opposite $\angle T$). This fits AAS.

Step6: Analyze Option 5

$AC = TQ = 3.2$cm and $CB = QP = 2.2$cm: Only sides congruent, no angles. This is SSA, not AAS/ASA.

Answer:

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