Chapter 7.1, Problem 4PSA

To determine

To calculate: Measure of $\angle M$

Answer to Problem 4PSA

Measure of $\angle M$ is $115°$ .

Explanation of Solution

Given information : Here, we have given that measure of $\angle G$ is $50°$

Formula used : Sum of all interior angles of a triangle is $180°$ .

Calculation: In $\angle GHJ$ ,

  $\begin{array}{l}\angle G+\angle H+\angle J=180°\\ \Rightarrow 50°+\angle H+\angle J=180°\\ \Rightarrow \angle H+\angle J=180°-50°\\ \Rightarrow \angle H+\angle J=130°\end{array}$

Now, diving both sides by 2, we get,

  $\begin{array}{l}\Rightarrow \frac{1}{2}(\angle H+\angle J)=\frac{1}{2}(130°)\\ \Rightarrow \frac{1}{2}\angle H+\frac{1}{2}\angle J=65°\end{array}$

Now, $\angle GHM=\angle MHJ$

  $\Rightarrow \angle MHJ=\frac{1}{2}\angle GHJ$

Similarly, $\angle GJM=\angle MJH$ ,

  $\Rightarrow \angle MJH=\frac{1}{2}\angle GJH$

Now, In $\Delta MHJ$ ,

  $\begin{array}{l}\angle M+\angle MHJ+\angle MJH=180°\\ \Rightarrow \angle M+\frac{1}{2}(\angle GHJ)+\frac{1}{2}(\angle GJH)=180°\\ \Rightarrow \angle M+65°=180°\\ \Rightarrow \angle M=180°-65°\\ \Rightarrow \angle M=115°\end{array}$