Converse Consecutive Interior Angles Theorem

The proof will be based on the given diagram, but it holds true for any pair of lines cut by a transversal. Consider only one pair of consecutive interior angles and one more angle.

It needs to be proven that l_1 and l_2 are parallel lines. It is already given that ∠ 1 is supplementary to ∠ 3. m∠ 1 + m∠ 3 = 180^(∘) The diagram shows that ∠ 3 and ∠ α form a linear pair, which means that these angles are supplementary. m∠ 3 + m∠ α = 180^(∘) Notice that ∠ 3 is common in both relationships. Subtracting the two equations gives a relation between ∠ 1 and ∠ α. m∠ 1 + m∠ 3 &= 180^(∘) ^- m∠ 3 + m∠ α &= 180^(∘) m∠ 1 - m∠ α &= 0^(∘) The last equation tells that ∠ 1 and ∠ α have the same measure and therefore, they are congruent. The diagram also shows that ∠ 1 and ∠ α are corresponding angles. Given that relation, the Converse Corresponding Angles Theorem can be applied.

Converse Corresponding Angles Theorem

If two lines are cut by a transversal so that the corresponding angles are congruent, then the lines are parallel.

Since ∠ 1 and ∠ α are corresponding congruent angles, then l_1 and l_2 are parallel lines. This completes the proof.