Exercise 18 Page 484

In the given diagram, we are told that $\angle ABD$ and $\angle CBE$ are the right angles.

We want to prove that $\angle ABC$ and $\angle DBE$ are congruent angles. To do so, we will start by completing the given two-column proof.

Two-Column Proof

Let's start by filling out the table from the first row that has an empty cell.

First row

The first statement describes an information that is already given. Thus, the first empty cell should be filled in with Given.

Third row

In the second row, the definition of complementary angles was used to determine that $\angle ABC$ and $\angle CBD$ are complementary. Notice that $\angle DBE$ and $\angle CBD$ together create $\angle CBE,$ which is a right angle. Therefore, using again that same definition, we can state that $\angle DBE$ and $\angle CBD$ are complementary angles as well.

Fourth row

According to the Congruent Complements Theorem, two angles that are complementary to the same angle, or to congruent angles, are congruent. In our case, since $\angle ABC$ is congruent to $\angle CBD$ and $\angle DBE$ is also congruent to $\angle CBD,$ we obtain that $\angle ABC\cong \angle DBE.$ With this information we can complete our two-column proof.

Flowchart

After completing our two-column proof, we can use similar steps to write a flowchart proof. Let's begin by reviewing the idea of a this type of proof. Arrows show the logical connections between the statements. Reasons are written below the statements. We can begin our proof with the two given pieces of information. This is that $\angle ABD$ and $\angle CBE$ are the right angles.

Now, we will follow similar steps as in the two-column proof. Here, we want to indicate that each piece of information leads to the conclusion about an appropriate pair of complementary angles. This is the reason why we divided the given information into two blocks.

Finally, using the Congruent Complements Theorem, we can connect the two newly created blocks. Using this connection, we can conclude that $\angle ABC$ and $\angle DBE$ are congruent angles.