1. **State the problem:** Given triangle $\triangle WXY$ with $m\angle X = 15^\circ$, $m\angle Y = 20^\circ$, and side $WY = 7.5$, find all sides and angles. 2. **Find the missing angle:** The sum of angles in a triangle is $180^\circ$. $$m\angle W = 180^\circ - m\angle X - m\angle Y = 180^\circ - 15^\circ - 20^\circ = 145^\circ$$ 3. **Label sides opposite to angles:** Side $WX$ is opposite $\angle Y$, side $XY$ is opposite $\angle W$, and side $WY$ is opposite $\angle X$. 4. **Use Law of Sines:** $$\frac{WX}{\sin 20^\circ} = \frac{XY}{\sin 145^\circ} = \frac{WY}{\sin 15^\circ}$$ 5. **Calculate the common ratio:** $$k = \frac{WY}{\sin 15^\circ} = \frac{7.5}{\sin 15^\circ}$$ Calculate $\sin 15^\circ \approx 0.2588$: $$k = \frac{7.5}{0.2588} \approx 28.96$$ 6. **Find side $WX$:** $$WX = k \times \sin 20^\circ = 28.96 \times 0.3420 \approx 9.90$$ 7. **Find side $XY$:** $$XY = k \times \sin 145^\circ = 28.96 \times \sin 145^\circ$$ Since $\sin 145^\circ = \sin (180^\circ - 145^\circ) = \sin 35^\circ \approx 0.574$$ $$XY = 28.96 \times 0.574 \approx 16.63$$ **Final answers:** - $m\angle W = 145^\circ$ - $WX \approx 9.90$ - $XY \approx 16.63$ - Given: $m\angle X = 15^\circ$, $m\angle Y = 20^\circ$, $WY = 7.5$