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} ≅ overline{pq}

∠a ≅ ∠t and ac = tq = 3.2cm

∠a ≅ ∠t and ∠b ≅ ∠p

∠a ≅ ∠t and overline{bc} ≅ overline{pq}

ac = tq = 3.2 cm and cb = qp = 2.2 cm

Explanation:

Step1: Recall AAS and ASA congruence criteria

• AAS (Angle - Angle - Side): Two angles and a non - included side of one triangle are congruent to two angles and the corresponding non - included side of another triangle.
• ASA (Angle - Side - Angle): Two angles and the included side of one triangle are congruent to two angles and the included side of another triangle.

Step2: Analyze each option

• Option 1: \(\angle B\cong\angle P\) and \(\overline{BC}\cong\overline{PQ}\)
• If we assume another pair of angles (say \(\angle C\) and \(\angle Q\)) are congruent (by triangle angle - sum property if we know enough information about the other angles), we can use AAS. But we need to check the correspondence. Let's assume we are comparing \(\triangle ABC\) and \(\triangle TPQ\). If \(\angle B\cong\angle P\), \(\overline{BC}\cong\overline{PQ}\), and if we can establish another angle (e.g., \(\angle A\cong\angle T\)) this can work for AAS.
• Option 2: \(\angle A\cong\angle T\) and \(AC = TQ=3.2\mathrm{cm}\)
• If we assume another pair of angles (say \(\angle C\) and \(\angle Q\)) are congruent (by triangle angle - sum property \(180^{\circ}-\angle A-\angle B = 180^{\circ}-\angle T-\angle P\) if \(\angle B\) and \(\angle P\) are related), we can use AAS. Also, if we consider the side - angle - angle, if we know the side is between two angles (but here we need to check correspondence. If \(\angle A\) and \(\angle T\) are angles and \(AC\) and \(TQ\) are sides, and if we can get another angle, it can be ASA or AAS.
• Option 3: \(\angle A\cong\angle T\) and \(\angle B\cong\angle P\)
• This is just two angles. We need a side for ASA or AAS. So this option alone is not sufficient.
• Option 4: \(\angle A\cong\angle T\) and \(\overline{BC}\cong\overline{PQ}\)
• For AAS: If \(\angle A\cong\angle T\) (angle), assume \(\angle C\cong\angle Q\) (by triangle angle sum \(180^{\circ}-\angle A - \angle B=180^{\circ}-\angle T-\angle P\) if \(\angle B\) and \(\angle P\) are related) and \(\overline{BC}\cong\overline{PQ}\) (non - included side), it satisfies AAS.
• Option 5: \(AC = TQ = 3.2\mathrm{cm}\) and \(CB=QP = 2.2\mathrm{cm}\)
• This is two sides. We need angles for ASA or AAS. So this option alone is not sufficient.

Answer:

\(\angle B\cong\angle P\) and \(\overline{BC}\cong\overline{PQ}\), \(\angle A\cong\angle T\) and \(AC = TQ = 3.2\mathrm{cm}\), \(\angle A\cong\angle T\) and \(\overline{BC}\cong\overline{PQ}\)