Question

if $\triangle rst \cong \triangle xyz$, which statement must be true?

a. $\overline{rs} \cong \overline{st}$

b. $\overline{st} \cong \overline{xz}$

c. $\angle r \cong \angle x$

d. $\angle s \cong \angle z$

Explanation:

Step1: Recall Congruent Triangles

When two triangles are congruent ($\triangle RST \cong \triangle XYZ$), corresponding parts (angles and sides) are congruent. The order of the vertices matters: $R$ corresponds to $X$, $S$ to $Y$, $T$ to $Z$.

Step2: Analyze Each Option

• Option A: $\overline{RS} \cong \overline{ST}$ – There's no info to say these sides in $\triangle RST$ are congruent. Eliminate.
• Option B: $\overline{ST} \cong \overline{XZ}$ – Corresponding side of $\overline{ST}$ should be $\overline{YZ}$ (since $S\to Y$, $T\to Z$), not $\overline{XZ}$. Eliminate.
• Option C: $\angle R \cong \angle X$ – $R$ corresponds to $X$ in congruent triangles, so their angles are congruent. This holds.
• Option D: $\angle S \cong \angle Z$ – $S$ corresponds to $Y$, so $\angle S$ should correspond to $\angle Y$, not $\angle Z$. Eliminate.

Answer:

C. $\angle R \cong \angle X$