Chapter 3.3, Problem 25WE

To determine

To prove: $\angle 1\cong \angle 2$

Explanation of Solution

Given:

The diagram:

  $\overline{BE}\perp \overline{DA}$

  $\overline{CD}\perp \overline{DA}$

Concept Used:

Theorem 3-7: In a plane two lines perpendicular to the same line are parallel.

Theorem 3-2: If two parallel lines are cut by a transversal then the alternate interior angles are congruent.

Consider the given figure:

It is given that, $\overline{BE}\perp \overline{DA}$ and $\overline{CD}\perp \overline{DA}$ , then using the theorem 3-7, it can be said that,

  $\overline{CD}\parallel \overline{BE}$

Now, lines $\overline{CD}\parallel \overline{BE}$ and $\overline{BD}$ is a transversal, so by Theorem 3-2,

  $\angle 1\cong \angle 2$ (Alternate interior angles)