1. **Stating the problem:** We have a right triangle XYZ with a right angle at Z, angle $\angle X = 69^\circ$, and side $YZ = 3$ (vertical side). We need to find the lengths $XY$, $YZ$, and the measure of angle $\angle Y$. 2. **Known information and formulas:** - Right triangle with $\angle Z = 90^\circ$. - $YZ = 3$ (given). - $\angle X = 69^\circ$ (given). - Sum of angles in a triangle is $180^\circ$. - Use trigonometric ratios: sine, cosine, tangent. 3. **Find $\angle Y$:** $$\angle Y = 180^\circ - 90^\circ - 69^\circ = 21^\circ$$ 4. **Label sides relative to $\angle X$:** - Opposite side to $\angle X$ is $YZ = 3$. - Adjacent side to $\angle X$ is $XY$. - Hypotenuse is $XZ$. 5. **Find hypotenuse $XZ$ using sine:** $$\sin(69^\circ) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{YZ}{XZ} = \frac{3}{XZ}$$ $$XZ = \frac{3}{\sin(69^\circ)}$$ Calculate $\sin(69^\circ) \approx 0.9336$: $$XZ = \frac{3}{0.9336} \approx 3.213$$ 6. **Find side $XY$ using cosine:** $$\cos(69^\circ) = \frac{XY}{XZ}$$ $$XY = XZ \times \cos(69^\circ)$$ Calculate $\cos(69^\circ) \approx 0.3584$: $$XY = 3.213 \times 0.3584 \approx 1.151$$ 7. **Final answers:** - $XY \approx 1.2$ - $YZ = 3$ - $m\angle Y = 21^\circ$