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 2 8 and 3 5.

**Proof.**

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

1 5.

Therefore, by the definition of congruent angles,

*m*1 = *m*5.

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

*m*1 + *m*2 = 180°.

Also, 5 and 8 are supplementary, so

*m*5 + *m*8 = 180°.

Substituting *m*1 for *m*5, we get

*m*1 + *m*8 = 180°.

Subtracting *m*1 from both sides, we have

*m*8 *= * 180° − *m*1 = *m*2.

Therefore, 2 8.

You can prove that 3 5 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*.