Question

1. jeff states that $\triangle pln \cong \triangle cvb$ because of asa. nancy says that she knows something that would allow her to use aas. what does she know that would allow her to use aas for these triangles?

Explanation:

Step1: Recall ASA and AAS

ASA (Angle - Side - Angle) requires two angles and the included side to be congruent. AAS (Angle - Angle - Side) requires two angles and a non - included side to be congruent.

Step2: Analyze Triangle Congruence

In triangle congruence, if we know that the sum of angles in a triangle is \(180^{\circ}\) (\(\angle A+\angle B+\angle C = 180^{\circ}\) for \(\triangle ABC\)). If Jeff used ASA, for Nancy to use AAS, she must know that the non - included side (corresponding to the AAS criterion) is congruent, or she knows that the third pair of angles are congruent (since if two angles are congruent, the third angle is also congruent because of the angle - sum property). But more precisely, in the context of triangle \(\triangle PLN\) and \(\triangle CVB\), if we assume some sides or angles are marked (from the diagram, we can infer that there are some congruent sides or angles). The key here is that in a triangle, if two angles are congruent, the third angle is congruent (because \(\angle1+\angle2+\angle3 = 180^{\circ}\), so if \(\angle1\cong\angle1'\) and \(\angle2\cong\angle2'\), then \(\angle3\cong\angle3'\)). Also, if we know that the sum of angles gives the third angle, and with a side (non - included for AAS), but more directly, the fact that the sum of interior angles of a triangle is \(180^{\circ}\) implies that if two angles are congruent, the third is too. However, the main thing Nancy knows is that the sum of the interior angles of a triangle is \(180^{\circ}\), so if two angles are congruent (from ASA - like information), the third angle is congruent, and then with a side (either the included side from ASA can be used as a non - included side for AAS if the third angle is known) or she knows that the triangles are triangles (so angle - sum property holds) and that the non - included side is congruent. But more simply, the key is that in a triangle, the sum of angles is \(180^{\circ}\), so if two angles are congruent, the third is congruent. So if Jeff has two angles and the included side (ASA), Nancy can use the angle - sum property to get the third angle congruent, and then use AAS (two angles and a non - included side). But the most direct answer is that she knows that the sum of the interior angles of a triangle is \(180^{\circ}\), so if two angles are congruent, the third angle is congruent, allowing her to use AAS (since AAS needs two angles and a non - included side, and with the angle - sum, she can get the third angle congruent). Alternatively, she knows that in a triangle, if two angles are congruent, the third angle is congruent (by the angle - sum property), so with the side (from ASA or another congruent side), she can apply AAS. The specific thing she knows is that the sum of the measures of the interior angles of a triangle is \(180^{\circ}\), so that if two angles are congruent, the third angle must also be congruent, which along with a side (either the included side from ASA used as a non - included side with the third angle or another congruent side) allows AAS. But more precisely, she knows that the triangles are triangles (so angle - sum property holds) and that the third pair of angles are congruent (because of the angle - sum property), so with a side (the side that is not included between the two angles used in ASA), she can use AAS. The main idea is that the sum of interior angles of a triangle is \(180^{\circ}\), so if two angles are congruent, the third is too, enabling AAS (which requires two angles and a non - included side; once the third angle is known to be congruent, we have two angles and a side).

Answer:

Nancy knows that the sum of the interior angles of a triangle is \(180^{\circ}\), so if two angles of one triangle are congruent to two angles of another triangle, the third angles must also be congruent (by the angle - sum property). This, along with a congruent side (either the included side from ASA can be used as a non - included side with the third angle or another congruent side), allows her to use AAS. Specifically, she knows that the third pair of angles are congruent (because of the angle - sum property of triangles), so with two angles and a non - included side, she can apply the AAS congruence criterion. (A more concise answer: She knows that the sum of interior angles of a triangle is \(180^{\circ}\), so the third angles are congruent, allowing AAS.)