1. **Stating the problem:** We are given two triangles, $\triangle XYZ$ and $\triangle RYS$, which are similar, denoted as $\triangle XYZ \sim \triangle RYS$. 2. **Understanding similarity:** When two triangles are similar, their corresponding angles are equal, and their corresponding sides are proportional. 3. **Corresponding angles:** Since $\triangle XYZ \sim \triangle RYS$, the angles correspond as follows: - $\angle X$ corresponds to $\angle R$ - $\angle Y$ corresponds to $\angle Y$ (common angle) - $\angle Z$ corresponds to $\angle S$ 4. **Corresponding sides:** The sides opposite these angles correspond: - Side $YZ$ corresponds to side $YS$ - Side $XZ$ corresponds to side $RS$ - Side $XY$ corresponds to side $RY$ 5. **Proportionality of sides:** The ratios of the lengths of corresponding sides are equal: $$\frac{XY}{RY} = \frac{YZ}{YS} = \frac{XZ}{RS}$$ 6. **Using the similarity:** This relationship allows us to find unknown side lengths or angle measures if some are given. **Final summary:** The similarity $\triangle XYZ \sim \triangle RYS$ means their corresponding angles are equal and their corresponding sides satisfy $$\frac{XY}{RY} = \frac{YZ}{YS} = \frac{XZ}{RS}$$, which is the key property to solve related problems involving these triangles.