Chapter 3.3, Problem 20WE

To determine

To explain: what can be said about the lines $\overline{PQ}$ and $\overline{RS}$ , it is given that $\angle 1\cong \angle 2$ and $\angle 4\cong \angle 5$ .

Answer to Problem 20WE

  $\overline{PQ}\parallel \overline{RS}$

Explanation of Solution

Given:

The diagram:

Concept Used:

Theorem 2-3: Vertically opposite angles are congruent.

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:

In this figure, by theorem 2-3,

  $\angle 2\cong \angle 5$ (Vertically opposite angles)

Also, it is given that,

  $\angle 1\cong \angle 2$ and $\angle 4\cong \angle 5$

Now applying the transitive property of congruence, it can be said that

  $\angle 1\cong \angle 4$

From the above figure, it can be observed that the angles $\angle 1$ and $\angle 4$ are the alternate interior angles which are formed when the lines $\overline{PQ}$ and $\overline{RS}$ are cut by the transversal $\overline{QR}$ .

And since $\angle 1\cong \angle 4$ , so by the theorem 3-5, it can be said that the lines $\overline{PQ}$ and $\overline{RS}$ are parallel.