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 , then and .
Proof.
Since , by the Corresponding Angles Postulate ,
.
Therefore, by the definition of congruent angles ,
.
Since and form a linear pair , they are supplementary , so
.
Also, and are supplementary, so
.
Substituting for , we get
.
Subtracting from both sides, we have
.
Therefore, .
You can prove that using the same method.
The converse of this theorem is also true; that is, if two lines and are cut by a transversal so that the alternate interior angles are congruent, then .