Question

4520va v. geometry sem a geometry
triangle congruence: asa and aas
which statements are true about additional information for proving that the
triangles are congruent? choose two correct answers.
if ( angle a cong angle t ), then the triangles would be
congruent by asa.
if ( overline { b c } cong overline { p q } ), then the triangles would be
congruent by asa.
if ( angle c ) and ( angle q ) are right angles, then triangles
would be congruent.
if ( angle b cong angle p ), then the triangles would be
congruent by aas.

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, 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, the triangles are congruent.

Step2: Analyze each option

• Option 1: If \(∠A\cong∠T\), then the triangles would be congruent by ASA

We know \(AC = QT\) (given by the tick mark). If \(∠A\cong∠T\) and \(∠C\) and \(∠Q\) (the angles marked with the same arc) are already congruent, but \(∠A\) and \(∠T\) are not the included angles for the side \(AC = QT\). So this is incorrect.

• Option 2: If \(\overline{BC}\cong\overline{PQ}\), then the triangles would be congruent by ASA

We have \(∠C\cong∠Q\) (marked angles) and \(AC = QT\) (tick - marked sides). If \(BC = PQ\), these are not the sides included between the two known congruent angles. So this is incorrect.

• Option 3: If \(∠C\) and \(∠Q\) are right angles, then triangles would be congruent

If \(∠C = ∠Q=90^{\circ}\), we still don't have enough information (we need another pair of angles or sides) to prove congruence. So this is incorrect.

• Option 4: If \(∠B\cong∠P\), then the triangles would be congruent by AAS

We know \(∠C\cong∠Q\) (marked angles) and \(AC = QT\) (tick - marked sides). If \(∠B\cong∠P\), by AAS (two angles and a non - included side), \(\triangle ABC\cong\triangle TPQ\).

Answer:

If \(∠B\cong∠P\), then the triangles would be congruent by AAS.