Exercise 19 Page 484

We are given a paragraph proof with several blanks and are asked to fill in those blank spaces. Then, we will also write a two-column proof.

Paragraph proof

Let's begin by looking at the given information and the desired outcome of the proof. Now, let's take a look at the statements that need to be completed one at a time.

First and Second Missing Information

Our first sentence containing blanks asks us what is the name of a pair of angles whose measures add up to $90^{\circ }.$ Since we know that the sum of the measures of complementary angles is $90^{\circ },$ we can fill the first blank with the word complementary. In the second blank we are supposed to write the sum of two angle measures which add to $90^{\circ }.$ From the given information we know that $\angle 1$ and $\angle 3$ are complementary angles. Therefore, the sum of their measures is $90^{\circ }.$ Therefore, we can fill the blank with $m\angle 1+m\angle 3.$

Third Missing Information

The next statement asks us about the name of the property used to write the equation $m\angle 1+m\angle 2$ $=$ $m\angle 1+m\angle 3.$ Since we have two expressions that are equal to $90^{\circ },$ these expressions must be equal to each other. This property is called Transitive Property of Equality.

Fourth Missing Information

The next statement asks us about the result of using the Subtraction Property of Equality. Let's perform a subtraction on both sides of the equation. Now, we can use this calculation to fill in the appropriate blank.

Fifth Missing Information

Our last statement is asking us to think about the relationship between angles which have the same measure. Angles that have the same measure are considered congruent. This is also the meaning of the symbol $(\cong )$ used in the conclusion. Let's fill in the last blank!

Completed Proof

$\angle 1$ and $\angle 2$ are complementary, and $\angle 1$ and $\angle 3$ are complementary. By the definition of complementary angles, $m\angle 1+m\angle 2=90^{\circ }$ and $\underline{m\angle 1+m\angle 3}=90^{\circ }.$ By the Transitive Property of Equality, $m\angle 1+m\angle 2=m\angle 1+m\angle 3.$ By the Subtraction Property of Equality, $\underline{m\angle 2=m\angle 3}.$ So, $\angle 2\cong \angle 3$ by the definition of congruent angles.

Two-column proof

Finally, we can summarize all steps in a two-column proof.

0. Statement	0. Reason
1. $\angle 1$ and $\angle 2$ are complementary. $\angle 1$ and $\angle 3$ are complementary.	1. Given
2. $m\angle 1+m\angle 2=90^{\circ }$ $m\angle 1+m\angle 3=90^{\circ }$	2. Definition of complementary angles
3. $m\angle 1+m\angle 2=m\angle 1+m\angle 3$	3. Transitive Property of Equality
4. $m\angle 2=m\angle 3$	4. Subtraction Property of Equality
5. $\angle 2\cong \angle 3$	5. Definition of congruent angles