Chapter 3.3, Problem 26WE

To determine

To prove: $\overline{CD}\perp \overline{DA}$

Explanation of Solution

Given:

The diagram:

  $\overline{BE}\perp \overline{DA}$

  $\angle C\cong \angle 3$

Concept Used:

Postulate 11: If two lines are cut by a transversal and corresponding angles are congruent, then the lines are parallel.

Theorem 3-4: If a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other one also.

Consider the given figure:

From the above figure, it can be observed that the angles $\angle C$ and $\angle 3$ are the corresponding angles which are formed when the lines $\overline{CD}$ and $\overline{BE}$ are cut by the transversal $\overline{CA}$ .

And since $\angle C\cong \angle 3$ , so by the postulate 11, it can be said that the lines $\overline{CD}$ and $\overline{BE}$ are parallel.

It is given that, $\overline{BE}\perp \overline{DA}$ and $\overline{CD}\parallel \overline{BE}$ , then using the theorem 3-4, it can be said that,

  $\overline{CD}\perp \overline{DA}$