Chapter 4.3, Problem 4CE

To determine

Toprove: $\overline{ZY}\parallel \overline{WX}$

Explanation of Solution

Given information: $\overline{ZW}\parallel \overline{YX}$ and $\overline{ZW}\cong \overline{XY}$

Proof:

  

Consider $\Delta WZX,\Delta YXZ$

  $\overline{ZW}\cong \overline{XY}$ (Given)

As $\overline{ZW}\parallel \overline{YX}$ , $\angle 3\cong \angle 4$

(if two lines are parallel then the alternate interior angles are equal. Here $\angle 3,\angle 4$ are the alternate interior angles)

  $\overline{ZX}\cong \overline{XZ}$ (common sides of $\Delta WZX,\Delta YXZ$ )

Therefore,

  $\Delta WZX\cong \Delta YXZ$ (by SAS congruence criteria)

Now if two triangles are congruent then their corresponding angles are also congruent.

As $\Delta WZX\cong \Delta YXZ$ ,

  $\angle 1\cong \angle 2$ (as $\angle 1,\angle 2$ are corresponding parts of $\Delta WZX,\Delta YXZ$ )

Now if alternate interior angles are equal then lines are said to be parallel.

As $\angle 1\cong \angle 2$ ,

  $\overline{ZY}\parallel \overline{WX}$