Let's begin by looking at the given information and the desired outcome of the proof. Then, we can write a paragraph proof!
&Given: ∠ 2 is supplementary to ∠ 7.
&Prove: l ∥ m
From the diagram, we can see that ∠ 2 and ∠ 3 form a linear pair. Recall that the Linear Pair Postulate tells us that if two angles form a linear pair, then they are supplementary. Therefore, by the Linear Pair Postulate, ∠ 2 and ∠ 6 are supplementary.
∠ 2 and ∠ 3 form a linear pair. Therefore, by the Linear Pair Postulate, ∠ 2 and ∠ 3 are supplementary.
Now we have the following statements.
∠ 2 and ∠ 7 are supplementary.
∠ 2 and ∠ 3 are supplementary.
The Congruent Supplements Theorem tells us that if two angles are supplements of the same angle, then the two angles are congruent. Thus, by the Congruent Supplements Theorem, ∠ 3 ≅ ∠ 7.
By the Congruent Supplements Theorem, ∠ 3 ≅ ∠ 7.
Now, we know that ∠ 3 and ∠ 7 are corresponding and congruent. The Converse of the Corresponding Angles Theorem tells us that if two lines and a transversal form corresponding angles that are congruent, then the lines are parallel. Therefore, by the Converse of the Corresponding Angles Theorem line l is parallel to line m.
∠ 3 and ∠ 7 are corresponding and congruent. By the Converse of the Corresponding Angles Theorem, l ∥ m.
Completed Proof
Considering the given information, we can summarize all the steps in a paragraph proof.
&Given: ∠ 2 is supplementary to ∠ 7.
&Prove: l ∥ m
Proof: ∠ 2 and ∠ 3 form a linear pair. Therefore, by the Linear Pair Postulate, ∠ 2 and ∠ 3 are supplementary. By the Congruent Supplements Theorem, ∠ 3 ≅ ∠ 7. ∠ 3 and ∠ 7 are corresponding and congruent. By the Converse of the Corresponding Angles Theorem, l ∥ m.
