Chapter 3.3, Problem 11ST1

To determine

To identify: all pairs of angles that must be congruent to $\angle 5$ given that $\overline{ED}\parallel \overline{AC}$ and $\overline{EB}\parallel \overline{DC}$ .

Answer to Problem 11ST1

  $\angle 2\text{ and }\angle 8$

Explanation of Solution

Given:

The diagram:

Concept Used:

Corresponding angles: These are the two angles which relative to the two lines are in the corresponding position.

Postulate 10: If two parallel lines are cut by a transversal then the corresponding angles are congruent.

Alternate interior angles: These are the two non adjacent interior angles which are on the opposite sides of the transversal.

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

Transitive Property: If $a=b$ and $b=c$ , then $a=c$ .

Consider the given figure:

In this figure, lines $\overline{EB}\parallel \overline{DC}$ and $\overline{BC}$ is a transversal, so by postulate 10,

  $\angle 5\cong \angle 2$ (Corresponding angles) ….. (1)

Similarly, lines $\overline{ED}\parallel \overline{AC}$ and $\overline{EB}$ is a transversal, so by theorem 3-2,

  $\angle 2\cong \angle 8$ (Alternate-interior angles)

Now, by using the transitive property on $\angle 5\cong \angle 2$ and $\angle 2\cong \angle 8$ .

Thus,

  $\angle 5\cong \angle 8$ ….. (2)