Question

Triangles RQS and NTV have the following characteristics:

• Right angles at ∠Q and ∠T
• RQ ≅ NT

Can it be concluded that ΔRQS ≅ ΔNTV by SAS? Why or why not?

Yes, one set of corresponding sides and one corresponding angle are congruent.
Yes, they are both right triangles.
No, it is necessary to know that another set of corresponding sides is congruent.
No, it is not possible for the triangles to be congruent.

Answer 1

To determine whether triangles \( RQS \) and \( NTV \) can be concluded to be congruent by the SAS (Side-Angle-Side) postulate, we need to analyze the information given about the triangles.

1. **Right angles at ∠Q and ∠T**: This means that both triangles have a right angle, which can be considered a corresponding angle. Since ∠Q ≅ ∠T (both being right angles), we have one pair of corresponding angles that are congruent.

2. **RQ ≅ NT**: This indicates that one set of corresponding sides is congruent.

Based on this information:
- We have one pair of sides that are congruent (RQ ≅ NT).
- We have one pair of angles that are congruent (∠Q ≅ ∠T).

For the triangles to be congruent by SAS, we would need two sides and the included angle between them to be congruent. In this case, we do not know the length of \( QS \) (side opposite to the angle ∠Q in triangle RQS) or \( TV \) (side opposite to the angle ∠T in triangle NTV).

Therefore, we cannot conclude that \( ΔRQS ≅ ΔNTV \) by SAS, because we lack information about another pair of corresponding sides.

**The correct response is:**
No, it is necessary to know that another set of corresponding sides is congruent.