Question

1. aas

a) \\( \angle r \cong \angle a \\)
b) \\( \overline { r s } \cong \overline { a s } \\)
c) \\( \overline { t r } \cong \overline { t a } \\)
d) \\( \overline { s t } \cong \overline { s t } \\)

Explanation:

Step1: Recall the AAS (Angle - Angle - Side) congruence criterion

AAS 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, then the two triangles are congruent.

Step2: Analyze each option

• Option A: If \(\angle R\cong\angle A\), along with the right angles (\(\angle ATS\) and \(\angle RTS\) are supplementary to the right angles at \(S\), and if we assume some other angle or side relations (since \(ST = ST\) (common side, reflexive property)), this can help in establishing AAS congruence.
• Option B: \(\overline{RS}\cong\overline{AS}\) is a side, but we need to check its position with respect to the angles. In AAS, the side is non - included. Just having this side congruent without knowing the angle - side - angle relation properly (in the context of AAS) is not sufficient.
• Option C: \(\overline{TR}\cong\overline{TA}\) is a side, but again, its relation to the angles for AAS is not clear. AAS requires two angles and a non - included side.
• Option D: \(\overline{ST}\cong\overline{ST}\) is the reflexive property (common side), but by itself (without the angle - angle - side combination where the side is non - included) it does not satisfy AAS.

Answer:

A. \(\angle R\cong\angle A\)