Alternate Interior Angles Theorem: Proof
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
- Two parallel lines are cut by a transversal. Find x and each angle.
- A transversal crosses two lines. Are they parallel?
- Find x for a co-interior pair measuring (3x)° and (2x + 30)°.