Question 1210462
Here's an analysis of the statements based on the fact that $\triangle RST \cong \triangle XYZ$ (congruent triangles).

Since the triangles are congruent, their **corresponding vertices, sides, and angles** are equal. The order of the letters defines the correspondence:
* $R \leftrightarrow X$
* $S \leftrightarrow Y$
* $T \leftrightarrow Z$

Therefore, the corresponding parts are:
* **Sides:** $RS = XY$, $ST = YZ$, $RT = XZ$
* **Angles:** $\angle RST = \angle XYZ$, $\angle STR = \angle YZX$, $\angle TRS = \angle ZXY$

Now let's evaluate each statement:

---

## 🧐 Evaluating the Statements

### (a) If RST is isosceles, then XYZ is isosceles.

**True.**
If $\triangle RST$ is isosceles, two of its sides are equal (e.g., $RS = RT$). Since the triangles are congruent, the corresponding sides are also equal ($RS = XY$ and $RT = XZ$). By substitution, $XY = XZ$. If two sides of $\triangle XYZ$ are equal, it is also **isosceles**.

### (b) If $ST = 6$, then $WY = 6$

**False.**
The corresponding side to $ST$ is **$YZ$**, so $YZ = 6$.
$W$ lies on $XY$, and $Y$ is a vertex. The length $WY$ is just a segment of the side $XY$. The length of $WY$ is generally not equal to $ST$.

### (c) If $RP \perp ST$, then $WY \perp YZ$

**False.**
* The line segment $RP$ is perpendicular to the side $ST$. This means $RP$ is an **altitude** of $\triangle RST$.
* The corresponding segment in $\triangle XYZ$ is $XW$ (since $R \leftrightarrow X$, $P \leftrightarrow W$, and $S \leftrightarrow Y$, $T \leftrightarrow Z$).
* The corresponding side to $ST$ is $YZ$.
* Therefore, the congruent statement is: **If $RP \perp ST$, then $XW \perp YZ$** (meaning $XW$ is an altitude of $\triangle XYZ$).
* The statement asks if $WY \perp YZ$. $WY$ is a segment of the side $XY$ and is generally **not** perpendicular to $YZ$.

### (d) If $RP$ bisects $\angle SRT$, then $XYZ$ is right.

**False.**
* If $RP$ bisects $\angle SRT$, it means $RP$ is an **angle bisector** of $\angle R$.
* Since $R \leftrightarrow X$, $XW$ is the corresponding angle bisector of $\angle X$ in $\triangle XYZ$.
* If the angle bisector of a triangle is also the altitude (from statement (c)), the triangle must be **isosceles**. This means $RS = RT$ and $XY = XZ$.
* $RP$ being an angle bisector does **not** make $\triangle RST$ a right triangle, and thus it does not make $\triangle XYZ$ a right triangle. $\triangle XYZ$ would be isosceles.

---

The only true statement is **(a)**.