Question

triangle congruence: asa and aas
which statements are true about additional information for proving that the triangles are congruent? choose two correct answers.
if all the angles are acute, then the triangles would be congruent.
if ∠c and ∠q are right angles, then triangles would be congruent.
if ∠b ≅ ∠p, then the triangles would be congruent by aas.
if ∠a ≅ ∠t, then the triangles would be congruent by asa.
if \\(\overline{bc} \cong \overline{pq}\\), then the triangles would be congruent by asa.

Explanation:

Step1: Analyze Triangle Congruence Conditions

First, recall ASA (Angle - Side - Angle) and AAS (Angle - Angle - Side) congruence criteria. ASA requires two angles and the included side to be congruent, while AAS requires two angles and a non - included side to be congruent.

Looking at the triangles \(\triangle ABC\) and \(\triangle TPQ\) (assuming the labels, we can see that there is a marked side and a marked angle in each triangle.

Step2: Evaluate Each Statement

• Statement 1: "If all the angles are acute, then the triangles would be congruent." Just having all angles acute does not imply congruence. Triangles can have all acute angles but different side lengths. So this is false.
• Statement 2: "If \(\angle C\) and \(\angle Q\) are right angles, then triangles would be congruent." Just being right - angled is not enough. We need more information about sides or other angles. So this is false.
• Statement 3: "If \(\angle B\cong\angle P\), then the triangles would be congruent by AAS." Let's check the given markings. We have one side marked congruent (the side with the tick mark) and if \(\angle B\cong\angle P\) and we can assume \(\angle C\cong\angle Q\) (from the angle markings), then we have two angles and a non - included side (the marked side) congruent, which is AAS. So this is true.
• Statement 4: "If \(\angle A\cong\angle T\), then the triangles would be congruent by ASA." If \(\angle A\cong\angle T\), and we have the included side (the marked side) and the other angle (from the angle markings), this satisfies ASA (two angles and the included side). So this is true.
• Statement 5: "If \(\overline{BC}\cong\overline{PQ}\), then the triangles would be congruent by ASA." \(\overline{BC}\) and \(\overline{PQ}\) are not the included sides for the given angles. So this does not satisfy ASA. So this is false.

Answer:

The two correct statements are: "If \(\angle B\cong\angle P\), then the triangles would be congruent by AAS" and "If \(\angle A\cong\angle T\), then the triangles would be congruent by ASA"