Question

Angles U and V are supplementary angles. The ratio of their measures is 7:13 Find the measure of each angle

Answer 1

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 $u:$

$u=180^{\circ}-v\\\\ u=180^{\circ}-117^{\circ}\\\\ \boxed{\begin{array}{c} u=63^{\circ} \end{array}}$