The Consecutive Interior Angles Theorem states that the consecutive interior angles on the same side of a transversal line intersecting two parallel lines are supplementary (That is, their sum adds up to 180).
Here we will prove its converse- if the consecutive interior angles on the same side of a transversal line intersecting two lines are supplementary, then the two lines are parallel.
Problem
m∠5 + m∠4 = 180°, prove that AB||CD
Strategy
This is a converse theorem, so we will look at what we did in the original theorem. Then, we’ll try to replicate that, switching what was given and what we need to prove.
In proving the original theorem, we relied on the fact that a linear pair of angles are supplementary. Let’s do that here, too: m∠1 + m∠4 = 180° as a linear pair, m∠5 + m∠4 = 180° is given, so m∠5=m∠1, and by the converse of the corresponding angles theorem, the lines are parallel.
Proof
(1) m∠5 + m∠4 = 180° // Given
(2) m∠1 + m∠4 = 180° // Linear pair of angles are supplementary
(3) m∠5=m∠1 // (1), (2) , transitive property of equality
(4) AB||CD // Converse of the corresponding angles theorem