Question

triangles rqs and ntv have the following characteristics:

• right angles at ∠q and ∠t
• $overline{rq} \cong \overline{nt}$

can it be concluded that $\triangle rqs \cong \triangle ntv$ by sas? why or why not?

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

Explanation:

To determine if \(\triangle RQS \cong \triangle NTV\) by SAS (Side - Angle - Side) congruence criterion, we recall that the SAS criterion requires two sides and the included angle of one triangle to be congruent to the corresponding two sides and the included angle of the other triangle.

We know that \(\angle Q\) and \(\angle T\) are right angles (so \(\angle Q\cong\angle T\)) and \(\overline{RQ}\cong\overline{NT}\). But for SAS, we need another pair of corresponding sides (the sides that form the included angle with the given side) to be congruent. In right - angled triangles \(\triangle RQS\) and \(\triangle NTV\), with right angles at \(Q\) and \(T\) respectively, and \(\overline{RQ}\cong\overline{NT}\), we need either \(\overline{QS}\cong\overline{TV}\) or \(\overline{RS}\cong\overline{NV}\) (but \(\overline{RS}\) and \(\overline{NV}\) are hypotenuses, and for SAS we need the sides adjacent to the right angle). Since we only know one pair of sides (\(\overline{RQ}\cong\overline{NT}\)) and one pair of angles (\(\angle Q\cong\angle T\)), we need another pair of corresponding sides to apply SAS.

• Option 1: SAS requires two sides and the included angle. One side and one angle are not enough for SAS, so this is incorrect.
• Option 2: Just being right - angled triangles does not imply congruence. There are many right - angled triangles with different side lengths that are not congruent, so this is incorrect.
• Option 3: This is correct because for SAS, we need two sides and the included angle. We have one side (\(\overline{RQ}\cong\overline{NT}\)) and the included angle (\(\angle Q\cong\angle T\)), but we need another pair of corresponding sides.
• Option 4: It is possible for the triangles to be congruent if the other pair of sides is congruent, so this is incorrect.

Answer:

C. No, it is necessary to know that another set of corresponding sides is congruent.