In
the drawing below, angles 3 and 6 are alternate interior angles,
as are angles 4 and 5. Alternate interior angles are congruent.
Formally, alternate interior angles are two interior
angles which lie on different parallel
lines and on opposite sides of
a
transversal.
Problem: Two parallel lines are cut by a transversal. One of the alternate interior angles measures 65°. What is the measure of the other alternate interior angle?
Step 1: Identify the angle pair. The two angles are between the parallel lines (interior) and on opposite sides of the transversal, so they are alternate interior angles.
Step 2: Apply the Alternate Interior Angles Theorem: when two parallel lines are cut by a transversal, alternate interior angles are congruent.
∠1≅∠2
Step 3: Since the first angle measures 65°, the alternate interior angle must also measure 65°.
∠2=65°
Answer: The other alternate interior angle measures 65°.
Another Example
Problem: Two parallel lines are cut by a transversal. One alternate interior angle is given as (3x + 10)° and the other as (5x − 20)°. Find the value of x and the measure of each angle.
Step 1: Because the lines are parallel, alternate interior angles are congruent. Set the two expressions equal.
3x+10=5x−20
Step 2: Solve for x. Subtract 3x from both sides, then add 20 to both sides.
10=2x−20⟹30=2x⟹x=15
Step 3: Substitute x = 15 back into either expression to find the angle measure.
3(15)+10=45+10=55°
Step 4: Verify with the other expression.
5(15)−20=75−20=55°✓
Answer: x = 15, and each alternate interior angle measures 55°.
Frequently Asked Questions
Are alternate interior angles always equal?
Alternate interior angles are equal (congruent) only when the two lines cut by the transversal are parallel. If the lines are not parallel, the angles still exist as a pair, but they will not be equal. In fact, this works in reverse too: if a transversal creates equal alternate interior angles, you can conclude the two lines must be parallel.
How do you tell alternate interior angles apart from consecutive (co-interior) angles?
Both types are interior angles (between the parallel lines). The key difference is their position relative to the transversal. Alternate interior angles are on opposite sides of the transversal and are congruent. Consecutive interior angles (also called co-interior or same-side interior angles) are on the same side and are supplementary, meaning they add up to 180°.
Alternate Interior Angles vs. Alternate Exterior Angles
Alternate interior angles lie between the two parallel lines, while alternate exterior angles lie outside them. They occupy different positions in the figure but obey the same congruence rule.
Why It Matters
Alternate interior angles are one of the fundamental tools for proving lines are parallel and for finding unknown angle measures in geometry. They appear constantly in proofs, construction problems, and real-world applications like engineering and architecture where parallel beams or edges are cut by diagonal supports. Understanding this angle relationship also builds toward more advanced topics like triangle angle sums and properties of parallelograms.
Common Mistakes
Mistake: Confusing alternate interior angles with consecutive (same-side) interior angles and assuming they are equal.
Correction: Alternate interior angles are on opposite sides of the transversal and are congruent. Consecutive interior angles are on the same side and are supplementary (sum to 180°). Always check which side of the transversal each angle is on.
Mistake: Assuming alternate interior angles are congruent even when the lines are not parallel.
Correction: The congruence relationship requires the two lines to be parallel. If you are not told the lines are parallel, you cannot assume the alternate interior angles are equal. Conversely, if you can show the angles are equal, that proves the lines are parallel.
