Question

a) ∠y≅∠t or ∠z≅∠u
b) (overline{xy}congoverline{st}) or (overline{zx}congoverline{us})
c) ∠x≅∠s or ∠y≅∠t
d) (overline{yz}congoverline{tu}) or (overline{zx}congoverline{us})

Explanation:

Step1: Recall triangle - congruence criteria

We know that for two triangles to be congruent, we have criteria like SSS (Side - Side - Side), SAS (Side - Angle - Side), ASA (Angle - Side - Angle), AAS (Angle - Angle - Side), and HL (Hypotenuse - Leg for right - triangles). Looking at the given triangles, we need to find the additional condition for congruence.

Step2: Analyze the marked parts

In \(\triangle XYZ\) and \(\triangle STU\), we have one side marked congruent (the red - marked sides) and one angle marked congruent (the red - marked angles). For SAS (Side - Angle - Side) congruence criterion, we need the sides including the marked angles to be congruent.
In \(\triangle XYZ\) and \(\triangle STU\), if \(\overline{XY}\cong\overline{ST}\) (the sides including the marked angles) and the marked angles \(\angle X\cong\angle S\) and the other marked sides are also considered in the congruence conditions, the triangles can be proved congruent. Similarly, if \(\overline{ZX}\cong\overline{US}\), along with the given angle and side congruences, we can prove congruence.

Answer:

B. \(\overline{XY}\cong\overline{ST}\) or \(\overline{ZX}\cong\overline{US}\)