Chapter 1.5, Problem 23PSC

To determine

To Calculate: The measure of angle DAE.

Answer to Problem 23PSC

  $m\angle DAE=60°$

Explanation of Solution

Given:

  

  $\begin{array}{l}\angle BAC=120°\\ \overrightarrow{AD}\text{ bisects }\angle BAF\\ \overrightarrow{AE}\text{ bisects }\angle CAF\end{array}$

Concept Used:

Bisectors divides the angle into two equal parts.

Calculation:

Here, we have

  $\begin{array}{l}\overrightarrow{AD}\text{ bisects }\angle BAF\\ \Rightarrow \angle BAD=\angle DAF\\ \overrightarrow{AE}\text{ bisects }\angle CAF\\ \Rightarrow \angle FAE=\angle EAC\\ \angle BAC=120°\\ \because \angle BAC=\angle BAD+\angle DAF+\angle FAE+\angle EAC\\ \Rightarrow 120°=2\angle DAF+2\angle FAE\\ \Rightarrow 120°=2\left(\angle DAF+\angle FAE\right)\\ \Rightarrow 60°=\left(\angle DAE\right)\\ \therefore m\angle DAE=60°\end{array}$

Conclusion:

  $m\angle DAE=60°$