Chapter 2.3, Problem 7ST1

To determine

The measure of $\angle EBC$

Answer to Problem 7ST1

The two other kinds of reasons that can be used in the proofs are postulates and definitions also the properties of algebra

Explanation of Solution

Given information:

The measure of $\angle ABC$ is ${108}^{\circ }.\overrightarrow{BD}$ is the bisector of $\angle ABC,$ and $\overrightarrow{BE}$ is the bisector of $\angle ABD.$

Consider that the measure of $\angle ABC$ is ${108}^{\circ }$ .

Also $\overrightarrow{BD}$ is the bisector of $\angle ABC$

Thus, by Angle Bisector Theorem,

  $\angle ABD=\frac{1}{2}\angle ABC$

And also $\overrightarrow{BE}$ is the bisector of $\angle ABD$

  $\angle EBC=\frac{1}{2}\angle ABD$

Thus

  $\begin{array}{l}\angle EBC=\frac{1}{2}\angle ABD\\ =\frac{1}{4}\angle ABC=\frac{1}{4}\times {108}^{\circ }={27}^{\circ }\end{array}$

Therefore, the measure of $\angle EBC$ is ${27}^{\circ }$