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 k || l, then angle2 is congruent to angle8 and angle3 is congruent to angle5.

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

Proof.

Since k || l, by the Corresponding Angles Postulate,

angle1 is congruent to angle5.

Therefore, by the definition of congruent angles,

mangle1 = mangle5.

Since angle1 and angle2 form a linear pair, they are supplementary, so

mangle1 + mangle2 = 180°.

Also, angle5 and angle8 are supplementary, so

mangle5 + mangle8 = 180°.

Substituting mangle1 for mangle5, we get

mangle1 + mangle8 = 180°.

Subtracting mangle1 from both sides, we have

mangle8 = 180° − mangle1 = mangle2.

Therefore, angle2 is congruent to angle8.

You can prove that angle3 is congruent to angle5 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 k || l.