# Alternate Interior Angles Theorem: Proof

## TL;DR

The alternate interior angles theorem states that when a transversal crosses two parallel lines, each pair of alternate interior angles is congruent (equal). This article gives the formal statement, a full two-step proof, the converse and its proof, the related co-interior angles theorem, and six worked examples. For the underlying definition of the angle pair, see Alternate Interior Angles.

## The Alternate Interior Angles Theorem — Statement

The **alternate interior angles theorem** states:

> **If a transversal crosses two parallel lines, then each pair of alternate interior angles is congruent (equal in measure).**

In the figure, lines lll and mmm are parallel and transversal ttt cuts them. The two alternate interior pairs are ∠3 and ∠5, and ∠4 and ∠6 — both pairs lie _between_ the parallel lines and on _opposite_ sides of the transversal. The theorem promises:

∠3 = ∠5 and ∠4 = ∠6.

The parallel condition is doing the work. Drop it — let the two lines tilt toward each other — and the alternate interior angles are generally unequal, and the theorem says nothing at all.

## How Do You Prove the Alternate Interior Angles Theorem?

The proof is short because it stands on two facts you already have about parallel lines: **corresponding angles are equal**, and **vertical angles are equal**. Chain them and the result drops out.

**Given:** Lines l∥m, cut by transversal t. Angles numbered ∠1 to ∠8 as in the figure.

**To prove:** ∠3 = ∠5 (one alternate interior pair).

| Step | Statement | Reason |
| --- | --- | --- |
| 1 | ∠3 = ∠1 | Vertical angles are equal (at the upper crossing) |
| 2 | ∠1 = ∠5 | Corresponding angles are equal (lines parallel) |
| 3 | ∠3 = ∠5 | Transitivity: both equal ∠1 |

The same two-step chain proves the other pair, ∠4 = ∠6. The theorem is proved.

## The Converse of the Alternate Interior Angles Theorem

The converse states:

> **If a transversal crosses two lines and a pair of alternate interior angles is congruent, then the two lines are parallel.**

**Proving the converse.** Suppose ∠3 = ∠5 (a pair of alternate interior angles is given equal), and we want to show l∥m.

| Step | Statement | Reason |
| --- | --- | --- |
| 1 | ∠3 = ∠1 | Vertical angles are equal |
| 2 | ∠3 = ∠5 | Given |
| 3 | ∠1 = ∠5 | From steps 1 and 2 |
| 4 | l∥m | Converse of the corresponding angles postulate |

## The Co-Interior Angles Theorem (the Supplementary Cousin)

The **co-interior angles theorem** states:

> **If a transversal crosses two parallel lines, each pair of co-interior angles is supplementary — they add to 180°.**

## Examples of Alternate Interior Angles Theorem

### **Example 1**
A transversal crosses two parallel lines. One alternate interior angle measures 72°. By the theorem, what does its alternate interior partner measure? 72°.

### **Example 2**
One alternate interior angle is (3x + 12)° and its partner is (5x − 18)°. Set 3x + 12 = 5x - 18, giving x = 15. The angles are 57°.

### **Example 3**
Angle vertically opposite to its partner is 96°. The alternate interior angle is also 96°. So, 4y=96 gives y=24.

### **Example 4**
If alternate interior angles measure 108°, then the lines are parallel by converse of the theorem.

### **Example 5**
A co-interior pair measures (2x + 20)° and (3x + 10)°. They add to 180°, which gives x = 30.

### **Example 6**
If ∠4 = ∠6, conclude the lines are parallel by the converse of the alternate interior angles theorem.

## Why the Theorem and Its Converse Matter Together

The theorem and its converse allow for practical applications in geometry and are essential in various proofs.

## Key Takeaways

- The **alternate interior angles theorem** states that congruent angles indicate parallel lines.
- Always confirm the parallel condition before applying the forward theorem.

## Practice These Problems to Solidify Your Understanding
1. Two parallel lines are cut by a transversal. Find x and each angle. 
2. A transversal crosses two lines. Are they parallel?
3. Find x for a co-interior pair measuring (3x)° and (2x + 30)°.