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