1. **Stating the problem:** We have a right triangle RST with a right angle at vertex S. Given: $t = 3$, where $t$ is the length of side RT. We need to find the angles $\angle R$, $\angle S$, and $\angle T$ and the lengths $s$ and $r$. 2. **Understanding the triangle:** Since $\angle S$ is a right angle, $\angle S = 90^\circ$. The sides are labeled as follows: - Hypotenuse: $s = RS$ - One leg: $r = ST$ - Other leg: $t = RT = 3$ 3. **Using the Pythagorean theorem:** $$s^2 = r^2 + t^2$$ 4. **Finding angles $\angle R$ and $\angle T$:** Using trigonometric ratios: - $\sin(\angle R) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{t}{s} = \frac{3}{s}$ - $\cos(\angle R) = \frac{r}{s}$ - $\tan(\angle R) = \frac{t}{r} = \frac{3}{r}$ Similarly for $\angle T$: - $\sin(\angle T) = \frac{r}{s}$ - $\cos(\angle T) = \frac{t}{s} = \frac{3}{s}$ - $\tan(\angle T) = \frac{r}{t} = \frac{r}{3}$ 5. **Since $r$ and $s$ are not given, we cannot find exact numeric values for $\angle R$ and $\angle T$ without more information.** 6. **Summary:** - $\angle S = 90^\circ$ - $\angle R$ and $\angle T$ depend on $r$ and $s$ which are unknown. If you provide values for $r$ or $s$, we can calculate the angles. **Final answers:** $$\angle S = 90^\circ$$ $$\angle R = \arctan\left(\frac{3}{r}\right)$$ $$\angle T = 90^\circ - \angle R$$