Let's begin with reviewing the idea of a two-column proof. It lists each statement on the left and its corresponding justification to its right. Each statement must follow logically from the previous steps. In this case, we are given that two consecutive interior angles are supplementary. Let's mark some angles in the example diagram.
We need to prove that k and j are parallel lines. Recall that we are given that ∠ 1 and ∠ 2 are supplementary. This is how we begin our proof!
Statement1)& ∠ 1and∠ 2are supplementary
Reason1)& Given
Let's now consider ∠ 1 and ∠ 3.
Note that these angles are adjacent. Additionally, their uncommon sides are opposite rays. Therefore, by the definition of a linear pair, ∠ 1 and ∠ 3 form a linear pair.
Statement2)& ∠ 1and∠ 3are linear pair
Reason2)& Definition of a linear pair
By the Linear Pair Postulate, ∠ 1 and ∠ 3 are supplementary angles.
Statement3)& ∠ 1and∠ 3are supplementary
Reason3)& Linear Pair Postulate
Note that ∠ 2 and ∠ 3 are both supplementary to ∠ 1. The Congruent Supplements Theorem states that if two angles are supplementary to the same angle, then they are congruent. Therefore, we have that ∠ 2 and ∠3 are congruent angles.
Statement4)& ∠ 2≅∠ 3
Reason4)& Congruent Supplements Theorem
Let's add this information to our diagram.
If two lines are cut by a transversal so the corresponding angles are congruent, then the lines are parallel.
Since ∠ 2 and ∠ 3 are corresponding congruent angles, by the Corresponding Angles Converse we can conclude that the lines are parallel.
Statement5)& k ∥ j
Reason5)& Corresponding Angles Converse
Finally, we can summarize all the steps is a two-column proof!
