Alternate Interior Angle Theorem

The Alternate Interior Angles Theorem states that, when two parallel lines are cut by a transversal, the resulting alternate interior angles are congruent.

So, in the figure below, if kl , then 28 and 35 .

Two parallel lines cut by a transversal n, with angles labeled 1 through 8

Proof.

Since kl , by the Corresponding Angles Postulate,

15 .

Therefore, by the definition of congruent angles,

m1=m5 .

Since 1 and 2 form a linear pair, they are supplementary, so

m1+m2=180° .

Also, 5 and 8 are supplementary, so

m5+m8=180° .

Substituting m1 for m5 , we get

m1+m8=180° .

Subtracting m1 from both sides, we have

m8=180°m1 =m2 .

Therefore, 28 .

You can prove that 35 using the same method.

The converse of this theorem is also true; that is, if two lines k and l are cut by a transversal so that the alternate interior angles are congruent, then kl .