Exercise 8 Page 482

Let's begin by looking at the given information and the desired outcome of the proof. Then, we can write a paragraph proof! By the definition of a right angle, $\angle 1$ measures $90^{\circ }.$

By the definition of a right angle, $m\angle 1=90^{\circ }.$

Now, let's examine the diagram.

From the diagram, we can see that $\angle 1$ and $\angle 2$ form a linear pair. By the Linear Pair Postulate, we know that $\angle 1$ and $\angle 2$ are supplementary angles. This means that their measures add to $180^{\circ }.$

Since $\angle 1$ and $\angle 2$ form a linear pair, by the Linear Pair Postulate, $\angle 1$ and $\angle 2$ are supplementary. Therefore, $m\angle 1+m\angle 2=180^{\circ }.$

By the Substitution Property of Equality, we can substitute $90^{\circ }$ for $m\angle 1$ in the above equation.

By the Substitution Property of Equality, $90^{\circ }+m\angle 2=180^{\circ }.$

Then, using the Subtraction Property of Equality, we can find the measure of $\angle 2.$ We conclude that $\angle 2$ measures $90^{\circ }.$

By the Subtraction Property of Equality, $m\angle 2=90^{\circ }.$

Since $m\angle 2=90^{\circ },$ by the definition of right angle, we know that $\angle 2$ is a right angle.

By the definition of right angle, $\angle 2$ is a right angle.

Completed Proof

Considering the given information, we can summarize all the steps in a paragraph proof. Proof. By the definition of a right angle $m\angle 1=90^{\circ }.$ Since $\angle 1$ and $\angle 2$ form a linear pair, by the Linear Pair Postulate they are supplementary angles. Then, by the definition of supplementary angles $m\angle 1+m\angle 2=180^{\circ }.$ By the Substitution Property of Equality $90^{\circ }+m\angle 2=180^{\circ }.$ Then $m\angle 2=90^{\circ }$ by the Subtraction Property of Equality. Finally $\angle 2$ is a right angle by the definition of a right angle.