Consecutive Interior Angles – Practice Questions, Solutions & Quick Tips

When you draw a transversal across two lines, a pair of angles pops up inside the region between the lines and on the same side of the transversal. Those are the consecutive interior angles – sometimes called co‑interior angles. If the two lines happen to be parallel, a neat little rule kicks in: the two angles add up to 180°, i.e., they’re supplementary.

That rule, known as the Consecutive Interior Angle Theorem, is the workhorse behind most geometry questions you’ll see in schoolbooks or competitive exams. Below we’ll walk through a few examples, sprinkle in a few “aha!” moments, and then give you some fresh practice problems to try on your own.

Example 1 – Solving for the variable

Imagine a transversal slicing two parallel lines. The two interior angles are expressed as (4x + 8)° and (16x + 12)°. Because the lines are parallel, we know the angles must sum to 180°.

So we write:

(4x + 8) + (16x + 12) = 180

Combine like terms, get 20x + 20 = 180, then 20x = 160, which means x = 8. Plugging the value back in gives us the actual angles: 4·8 + 8 = 40° and 16·8 + 12 = 140°. Easy, right? Those two numbers, 40° and 140°, are indeed supplementary.

Example 2 – Checking parallelism

Sometimes the question flips: you’re given two interior angles, say 85° and 110°, and asked whether the lines are parallel. Add them up: 85 + 110 = 195°, which is not 180°. Since they’re not supplementary, the lines cannot be parallel. The theorem works both ways – it’s a quick litmus test.

Example 3 – Filling in missing angles

Let’s say we know one angle, ∠4, measures 65°. Because ∠4 and ∠6 are corresponding, ∠6 is also 65°. Now, interior angles on the same side of the transversal must sum to 180°, so ∠5 = 180 − 65 = 115°. Finally, ∠3 equals ∠6 (they’re opposite interior angles), giving us ∠3 = 115° as well.

Practice Problems

Problem 1: Two parallel lines are cut by a transversal. One interior angle is (2x − 7)°, the other is (x + 1)°. Find both angles.

Problem 2: On a set of parallel lines, ∠Q is 60° and it sits opposite a consecutive interior angle ∠P. What is ∠P?

Problem 3: The sum of two consecutive interior angles is (3z − 8)°, and one of them equals z degrees. Determine the measures of both angles.

Give these a go, and you’ll see how the theorem simply boils down to “add up to 180°”. If you get stuck, just remember: parallel lines ⇒ supplementary interior angles. That little fact will save you a lot of algebraic headache.

Happy solving, and keep an eye out for those “same‑side” angles the next time you sketch a transversal!