Exercise 20 Page 118

We are given the following diagram.

As we can see, there are two intersecting segments that form four angles. Let's first analyze the segments and then we will consider the angles.

Segments

On the diagram, two segments intersect each other, and we can see two red marks on $\overline{ED}.$

This means that $\overline{BD}$ and $\overline{EB}$ are congruent segments. Therefore, $\overline{AC}$ bisects $\overline{DE}.$

Angles

Analyzing the diagram, we can see that there are two pairs of vertical angles — $\angle 1$ and $\angle CBD,$ as well as $\angle 2$ and $\angle ABD.$ By the Vertical Angles Theorem, we can conclude that the angles in each pair are congruent. Therefore, the following statements are true. Also, there are four pairs of angles that form linear pairs in the diagram. Let's recall the Linear Pair Postulate.

Linear Pair Postulate

If two angles form a linear pair, then they are supplementary.

According to this postulate, the angles from each pair are supplementary angles.

	Supplementary Angles
Pair $1$	$\angle 1$ and $\angle 2$
Pair $2$	$\angle 2$ and $\angle CBD$
Pair $3$	$\angle CBD$ and $\angle ABD$
Pair $4$	$\angle ABD$ and $\angle 1$

Conclusion

Let's list all the statements that we can make based on the given diagram.

1. $\overline{BD}$ and $\overline{EB}$ are congruent segments.
2. $\overline{AC}$ bisects $\overline{DE}.$
3. $\angle 1$ and $\angle 2,$ $\angle 2$ and $\angle CBD,$ $\angle CBD$ and $\angle ABD,$ and $\angle ABD$ and $\angle 1$ form linear pairs.
4. $\angle 1$ and $\angle 2,$ $\angle 2$ and $\angle CBD,$ $\angle CBD$ and $\angle ABD,$ and $\angle ABD$ and $\angle 1$ are supplementary angles.
5. $\angle 1$ and $\angle CBD,$ as well as $\angle 2$ and $\angle ABD,$ are vertical angles.
6. $\angle 1$ and $\angle CBD,$ as well as $\angle 2$ and $\angle ABD,$ are are congruent angles.