Chapter 3.3, Problem 21WE

To determine

To explain: what can be said about the other angles, if it is given that $\angle 3\cong \angle 6$ .

Answer to Problem 21WE

  $\angle 2\cong \angle 5$

  $\angle 1\cong \angle 4$

Explanation of Solution

Given:

The diagram:

Concept Used:

Theorem 3-5: If two lines are cut by a transversal and alternate interior angles are congruent, then the lines are parallel.

Theorem 2-3: Vertically opposite angles are congruent.

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

Consider the given figure:

In the above figure, it can be observed that the angles $\angle 3$ and $\angle 6$ are the alternate interior angles which are formed when the lines $\overline{PQ}$ and $\overline{RS}$ are cut by the transversal $\overline{PS}$ .

And since $\angle 3\cong \angle 6$ , so by the theorem 3-5, it can be said that the lines $\overline{PQ}$ and $\overline{RS}$ are parallel.

Now, lines $\overline{PQ}\parallel \overline{RS}$ and $\overline{QR}$ is a transversal, so by Theorem 3-2,

  $\angle 1\cong \angle 4$ (Alternate interior angles)

Also in the above figure, by using theorem 2-3 it can be said that,

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