If \(u\) and \(v\) are supplementary angles, then
\(u+v=180^{\circ}~~~~~\mathbf{(i)}\)
The ratio of \(u\) and \(v\) is \(\dfrac{7}{13}:\)
\(\dfrac{u}{v}=\dfrac{7}{13}\\\\\\ u=\dfrac{7v}{13}~~~~\mathbf{(ii)}\)
Substitute \(\mathbf{(ii)}\) into \(\mathbf{(i)}:\)
\(\dfrac{7v}{13}+v=180^{\circ}\\\\\\ \left(\dfrac{7}{13}+1 \right )\cdot v=180^{\circ}\\\\\\ \left(\dfrac{7}{13}+\dfrac{13}{13} \right )\cdot v=180^{\circ}\\\\\\ \dfrac{20}{13}\cdot v=180^{\circ}\\\\\\ v=180^{\circ}\cdot \dfrac{13}{20}\\\\\\ \boxed{\begin{array}{c} v=117^{\circ} \end{array}}\)
From \(\mathbf{(i)},\) we find the measure of \(u:\)
\(u=180^{\circ}-v\\\\ u=180^{\circ}-117^{\circ}\\\\ \boxed{\begin{array}{c} u=63^{\circ} \end{array}}\)
