1. **State the problem:** We need to find the length of the missing side $w$ in a quadrilateral where the sides and angles are given as follows: one side is 20 m, and the angles adjacent to this side are 63° and 109°, while the angles adjacent to side $w$ are 37° and 49°. 2. **Identify the approach:** Since the quadrilateral has four angles summing to 360°, and we know all four angles (63°, 109°, 37°, 49°), we can use the Law of Sines in the triangles formed by the diagonal connecting the vertices opposite sides 20 m and $w$. 3. **Calculate the diagonal length:** The diagonal opposite to sides 20 m and $w$ can be found using the Law of Sines in one triangle. 4. **Apply Law of Sines in the triangle with side 20 m:** $$\frac{20}{\sin(37^\circ + 49^\circ)} = \frac{d}{\sin 63^\circ}$$ Calculate $\sin(37^\circ + 49^\circ) = \sin 86^\circ$. 5. **Calculate $d$:** $$d = \frac{20 \times \sin 63^\circ}{\sin 86^\circ}$$ 6. **Apply Law of Sines in the triangle with side $w$:** $$\frac{w}{\sin 109^\circ} = \frac{d}{\sin 49^\circ}$$ 7. **Substitute $d$ from step 5:** $$w = \frac{d \times \sin 109^\circ}{\sin 49^\circ} = \frac{20 \times \sin 63^\circ \times \sin 109^\circ}{\sin 86^\circ \times \sin 49^\circ}$$ 8. **Calculate the sines:** $\sin 63^\circ \approx 0.8910$ $\sin 86^\circ \approx 0.9986$ $\sin 109^\circ \approx 0.9455$ $\sin 49^\circ \approx 0.7547$ 9. **Calculate $w$ numerically:** $$w = \frac{20 \times 0.8910 \times 0.9455}{0.9986 \times 0.7547} \approx \frac{16.85}{0.7546} \approx 22.32$$ 10. **Final answer:** Rounded to the nearest integer, the length of side $w$ is **22** meters.