Chapter 3.3, Problem 24WE

To determine

To prove: $\overline{CD}\parallel \overline{BE}$

Explanation of Solution

Given:

The diagram:

The line $\overline{BE}$ bisects $\angle DBA$

  $\angle 3\cong \angle 1$

Concept Used:

Transitive property of congruence: if $a\cong b$ and $b\cong c$ , then $a\cong c$ .

Theorem 3-5: If two lines are cut by a transversal and alternate interior angles are congruent, then the lines are parallel.

Consider the given figure:

It is given that, the line $\overline{BE}$ bisects $\angle DBA$ , this implies that,

  $\begin{array}{c}m\angle DBE=m\angle EBA\\ m\angle 2=m\angle 3\end{array}$

That is,

  $\angle 2\cong \angle 3$

Now applying the transitive property of congruence on $\angle 1\cong \angle 3$ and $\angle 3\cong \angle 2$ , it can be said that

  $\angle 1\cong \angle 2$

From the above figure, it can be observed that the angles $\angle 1$ and $\angle 2$ are the alternate interior angles which are formed when the lines $\overline{CD}$ and $\overline{BE}$ are cut by the transversal $\overline{DB}$ .

And since $\angle 1\cong \angle 2$ , so by the theorem 3-5, it can be said that the lines $\overline{CD}$ and $\overline{BE}$ are parallel.