Question

triangles rqs and ntv have the following characteristics:

• right angles at \\(\angle q\\) and \\(\angle t\\)
• \\(\overline{rq} \cong \overline{nt}\\)

can it be concluded that \\(\delta rqs \cong \delta ntv\\) 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.

Explanation:

Analyze the given geometric information

Using the Right Triangle Definition knowledge point

$$ \angle Q = 90^\circ, \quad \angle T = 90^\circ \implies \angle Q \cong \angle T $$

$$ \overline{RQ} \cong \overline{NT} $$

Evaluate the SAS Congruence Postulate requirements

Using the SAS Congruence Postulate knowledge point

$$ \text{SAS requires: Side-Angle-Side} $$

$$ \text{Known: } \overline{RQ} \cong \overline{NT} \text{ (Side), } \angle Q \cong \angle T \text{ (Angle)} $$

$$ \text{Missing: } \overline{QS} \cong \overline{TV} \text{ (Side)} $$

Determine the correct conclusion

Since we only have one congruent side pair and one congruent angle pair, we cannot conclude congruence by SAS without knowing that the other adjacent side pair is congruent.

Answer:

• 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. (Correct answer)
• No, it is not possible for the triangles to be congruent.