Question

4520va v. geometry sem a geometry
triangle congruence: asa and aas
what additional information could be used to prove that the triangles are
congruent using aas or asa? choose three correct answers.
ac = tq = 3.2 cm and cb = qp = 2.2 cm
∠a ≅ ∠t and overline{bc} ≅ overline{pq}
∠a ≅ ∠t and ∠b ≅ ∠p
∠a ≅ ∠t and ac = tq = 3.2 cm
∠b ≅ ∠p and overline{bc} ≅ overline{pq}

Explanation:

Step1: Recall ASA and AAS congruence criteria

ASA (Angle - Side - Angle) states that if two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent. AAS (Angle - Angle - Side) states that if two angles and a non - included side of one triangle are congruent to two angles and the corresponding non - included side of another triangle, then the triangles are congruent.

Step2: Analyze each option

• Option 1: \(AC = TQ=3.2\mathrm{cm}\) and \(CB = QP = 2.2\mathrm{cm}\)

This is just two pairs of sides. We need angles for ASA or AAS. So this option is incorrect.

• Option 2: \(\angle A\cong\angle T\) and \(\overline{BC}\cong\overline{PQ}\)

If we assume \(\angle C\cong\angle Q\) (by vertical or other angle relations, not given in the problem, but if we consider the general case of triangle congruence set - up), with \(\angle A\cong\angle T\) (angle), \(\overline{BC}\cong\overline{PQ}\) (side) and if we can get another angle (say \(\angle B\cong\angle P\) which can be derived from triangle angle - sum property if two angles are known), this can be used for AAS.

• Option 3: \(\angle A\cong\angle T\) and \(\angle B\cong\angle P\)

If two angles (\(\angle A\cong\angle T\) and \(\angle B\cong\angle P\)) are congruent, then by the triangle angle - sum property (\(\angle A+\angle B+\angle C=180^{\circ}\) and \(\angle T+\angle P+\angle Q = 180^{\circ}\)), \(\angle C\cong\angle Q\). If we consider a side (by the order of ASA or AAS, assume a corresponding side), this can be used for ASA or AAS.

• Option 4: \(\angle A\cong\angle T\) and \(AC = TQ=3.2\mathrm{cm}\)

If \(\angle A\cong\angle T\) (angle) and \(AC = TQ\) (side), and if we can get another angle (say \(\angle C\cong\angle Q\) or \(\angle B\cong\angle P\)), this can be used for ASA or AAS.

• Option 5: \(\angle B\cong\angle P\) and \(\overline{BC}\cong\overline{PQ}\)

If \(\angle B\cong\angle P\) (angle) and \(\overline{BC}\cong\overline{PQ}\) (side), and if we can get another angle (say \(\angle C\cong\angle Q\) or \(\angle A\cong\angle T\)), this can be used for ASA or AAS.

Answer:

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