Exercise 3 Page 481

We are asked to rewrite the given two-column proof.

Statement	Reason
$\angle 5$ and $\angle 7$ are vertical angles	Given
$\angle 5$ and $\angle 6$ are a linear pair. $\angle 6$ and $\angle 7$ are a linear pair.	Definition of linear pair as shown in the diagram
$\angle 5$ and $\angle 6$ are supplementary. $\angle 6$ and $\angle 7$ are supplementary.	Linear Pair Postulate
$\angle 5\cong \angle 7$	Congruent Supplements Theorem

In the final step, the Congruent Supplements Theorem is used. It tells us that if two angles are supplementary to the same angle, then these angles are congruent.

Note that the proof corresponds to the diagram given in an example.

We want to rewrite the given proof without using the Congruent Supplements Theorem. We will use the first three steps of the given proof to show that $\angle 5\cong \angle 7.$

Rewriting the Proof

Since we cannot use the theorem, we will possibly need a few more steps to prove this statement. From step three, we know that both pairs of angles $\angle 5\&\angle 6$ and $\angle 6\&\angle 7$ are supplementary. By the definition of supplementary angles, we obtain that the measures of each pair add up to $180^{\circ }.$ By the Transitive Property of Equality, we can equate these two sums. This equation is our fifth statement. Notice that the term $m\angle 6$ is on both sides of the obtained equation. We can subtract this term from both sides by using the Subtraction Property of Equality. Therefore, our sixth statement is the above equation. We obtained that the measure of $\angle 5$ is equal to the measure of $\angle 7.$ By the definition of congruent angles, these two angles are congruent. This is our final statement.

Complete Proof

Finally, we can complete our two-column table after rewriting the proof. Recall that the first three steps are exactly the same for both proofs.

Statement	Reason
$\angle 5$ and $\angle 7$ are vertical angles	Given
$\angle 5$ and $\angle 6$ are a linear pair. $\angle 6$ and $\angle 7$ are a linear pair.	Definition of linear pair as shown in the diagram
$\angle 5$ and $\angle 6$ are supplementary. $\angle 6$ and $\angle 7$ are supplementary.	Linear Pair Postulate
$m\angle 5+m\angle 6=180^{\circ }$ $m\angle 6+m\angle 7=180^{\circ }$	Definition of Supplementary Angles
$m\angle 5+m\angle 6=m\angle 6+m\angle 7$	Substitution Property of Equality
$m\angle 5=m\angle 7$	Subtraction Property of Equality
$\angle 5\cong \angle 7$	Definition of Congruent Angles

In comparison with the given proof, we needed $3$ more steps to prove the statement without using the Congruent Supplements Theorem.