Definition: When a transversal (any line that passes through two other lines) cuts two coplanar lines then the angles that are in between the two coplanar lines and that lie on the opposite sides of the transversal are called the Alternate Interior angles.
Alternate Interior Angles

In these figures, Line m and line n are coplanar lines. The transversal cuts both the lines. $\angle$a and $\angle$c are angles formed when the transversal cuts m and n and these are one pair of angles that are on opposite sides of transversal l and in between the lines m and n. Thus $\angle$a and $\angle$c are Alternate Interior angles. There is one other such pair of Alternate Interior angles in these figures. $\angle$b and $\angle$d is the other pair of Alternate Interior angles in these figures. In Figure (i), the lines m and n are parallel and in Figure (ii) the lines m and n are not parallel.

It is easy to remember that the pair of Alternate Interior angles are on "Alternate" sides of the Transversal, and they are on the "Interior" of the two crossed lines.

If two parallel lines are cut by a transversal, then the pairs of Alternate Interior angles are always congruent.

Proof: To prove the theorem we need to prove that, if two lines M and N are cut by the transversal L, and if MN, then ∠1≅ ∠5 and ∠4≅ ∠8.

Alternate Interior Angles Theorem

From the above figure, Since M || N, by the Corresponding Angles Postulate,

We have ∠2≅ ∠8

Therefore, by the definition of congruent angles, ∠2= ∠8.

Since ∠1 and ∠2 form a linear pair, they are supplementary and so ∠1 + ∠2 = 180°.

Also, since ∠ 8 and ∠ 5 form a linear pair, they are supplementary and so ∠8 + ∠5 = 180°.

Substituting ∠2 for ∠8 (since ∠2= ∠8),

We get ∠2 + ∠5 = 180°.

Subtracting ∠2 from both sides of above equation, we get ∠5 = 180° - ∠2.

But 180° - ∠2 is ∠1.

Thus, ∠5 = ∠1.

Therefore, ∠1 is congruent to ∠5.

Similarly, it can be proved that ∠4 is congruent to ∠8 using the above same method.