Question

The figure is not drawn to scale. MN , PQ are straight lines. If

Answer 1

`\begin{gathered} \text{Given:} \\ m\angle p=4m\angle r,\text{ (}\angle p\text{ is four times as large as }\angle r\text{)} \\ m\angle q=54\degree,\text{ by vertical angle theorem. the angle opposite to the vertex of }\angle q \\ \text{Solve for }m\angle p\colon\text{ Since }\overline{MN}\text{ is a straight line, we can say that} \\ m\angle p+54\degree=180\degree \\ m\angle p=180\degree-54\degree \\ m\angle p=126\degree \\ \\ \text{Now that we have the value for }m\angle p,\text{ we can use it to solve for }m\angle r\text{.} \\ m\angle p=4m\angle r \\ (m\angle p)/(4)=\frac{\cancel{4}m\angle r}{\cancel{4}} \\ (m\angle p)/(4)=m\angle r\Rightarrow m\angle r=(m\angle p)/(4) \\ m\angle r=(m\angle p)/(4) \\ m\angle r=(126\degree)/(4) \\ m\angle r=31.5\degree \\ \\ \text{Now that we have both values for }m\angle q\text{ and }m\angle r,\text{ substitute it} \\ m\angle q+m\angle r=\text{?} \\ 54\degree+31.5\degree=85.5\degree \\ m\angle q+m\angle r=85.5\degree\text{ (final answer)} \end{gathered}`