Rule

Consecutive Interior Angles Theorem

If two parallel lines are cut by a transversal, then the consecutive interior angles are supplementary angles.
Two parallel lines cut by a transversal
In the diagram, ∠ 1 and ∠ 3 are supplementary as well as ∠ 2 and ∠ 4.


l_1 ∥ l_2 ⇒ m∠ 1 + m∠ 3 &= 180 ^(∘) m∠ 2 + m∠ 4 &= 180 ^(∘)

Proof

Start by noticing that ∠ 1 and ∠ 4 form a linear pair as well as ∠ 2 and ∠ 3.

Two parallel lines cut by a transversal
Therefore, these pair of angles are supplementary. m∠ 1 + m∠ 4 = 180^(∘) & (I) m∠ 2 + m∠ 3 = 180^(∘) & (II) On the other hand, since the lines l_1 and l_2 are parallel, by the Alternate Interior Angles Theorem, the alternate interior angles are congruent. This allows to write the following relations. ∠ 1 ≅ ∠ 2 & ⇒ m∠ 1 = m∠ 2 [0.25em] ∠ 3 ≅ ∠ 4 & ⇒ m∠ 3 = m∠ 4 Next, in Equation I, substitute m∠ 3 for m∠ 4.
m∠ 1 + m∠ 4 = 180^(∘)
m∠ 1 + m∠ 3 = 180^(∘)
The last equation implies that ∠ 1 and ∠ 3 are supplementary. Similarly, in Equation II, substitute m∠ 4 for m∠ 3.
m∠ 2 + m∠ 3 = 180^(∘)
m∠ 2 + m∠ 4 = 180^(∘)
As before, the last equation implies that ∠ 2 and ∠ 4 are supplementary, which completes the proof.
Exercises