Consecutive Interior Angles Theorem

The Theorem

The two interior angles formed by a transversal line intersecting two parallel lines are supplementary (i.e: sum up to 180°).

The problem

AB||CD, prove m∠5 + m∠4 = 180°

So how do we go about this? We already know that the 2 angles that are next to each other and which form a straight line are “Supplementary angles” and their sum is 180°. So we will try to use that since here we also need to prove that the sum of two angles is  180°.

So let’s proceed to the proof, using what we already know about angles that are next to each other and which form a straight line.

Proof

(1)    AB||CD                                //given

(2)    ∠1 ≅ ∠5                               //from the axiom of parallel lines – corresponding angles

(3)    m∠1 = m∠5                       //definition of congruent angles 

(4)    m∠1 + m∠4 = 180°           // straight line measures 180° 

(5)    m∠5 + m∠4 = 180°           //using (3) and performing algebraic substitution, replacing  m∠1 with the equivalent m∠5

And we have proven the theorem.