Chapter 1.5, Problem 10PSA

To determine

To Prove: $\overrightarrow{EG}\text{bisects }\angle DEF$ .

Explanation of Solution

Given:

  

  $\angle DEG\cong \angle FEG$

If a ray divides an angle into two angles, the ray bisects the angle.

Concept Used:

Bisector of angle divides it into two equal parts.

Proof:

Here, we have

  $\begin{array}{l}\angle DEG\cong \angle FEG\\ \Rightarrow m\angle DEG=m\angle FEG\\ \text{If two angles having one arm in common are equal in measure, then the common arm}\\ \text{will be the bisector of larger angle, i}\text{.e}\text{., }\\ \text{a ray which divides the angle into two equal parts is the bisector of the angle, }\\ \text{hence bisect the angle}\text{. }\\ \text{Thus, here the ray }\overrightarrow{EG}\text{bisects }\angle DEF\text{ and thereby the bisector of the angle }\angle DEF.\\ \Rightarrow \overrightarrow{EG}\text{bisects }\angle DEF\end{array}$