Chapter 4.3, Problem 3CE

To determine

To prove: $\overline{WX}\parallel \overline{ZY}$

Explanation of Solution

Given information: $\overline{WX}\cong \overline{YZ}$ and $\overline{ZW}\cong \overline{XY}$

Proof:

  

Consider $\Delta WZX,\Delta YXZ$

  $\overline{WX}\cong \overline{YZ}$ (Given)

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

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

Therefore,

  $\Delta WZX\cong \Delta YXZ$ (by SSS 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 $\angle 1\cong \angle 2$ ,

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