In the figure
below, angles 4 and 6 are consecutive interior angles. So are angles
3 and 5.
Consecutive interior angles are supplementary. Formally, consecutive
interior angles may be defined as two interior
angles lying on
the same side of the transversal cutting across two parallel
lines.
∠B = The other consecutive interior angle on the same side of the transversal
Worked Example
Problem: Two parallel lines are cut by a transversal. One of the consecutive interior angles measures 65°. Find the measure of the other consecutive interior angle.
Step 1: Identify the relationship. Consecutive interior angles formed by a transversal crossing parallel lines are supplementary, so they add up to 180°.
∠A+∠B=180°
Step 2: Substitute the known angle into the equation.
65°+∠B=180°
Step 3: Solve for the unknown angle by subtracting 65° from both sides.
∠B=180°−65°=115°
Answer: The other consecutive interior angle measures 115°.
Another Example
Problem: Two parallel lines are cut by a transversal. One consecutive interior angle is given as (3x + 10)° and the other as (2x + 20)°. Find the value of x and both angle measures.
Step 1: Since the lines are parallel, consecutive interior angles are supplementary.
(3x+10)+(2x+20)=180
Step 2: Combine like terms on the left side.
5x+30=180
Step 3: Solve for x by subtracting 30 from both sides, then dividing by 5.
5x=150⇒x=30
Step 4: Substitute x = 30 back into each expression to find the angle measures.
3(30)+10=100°2(30)+20=80°
Answer: x = 30. The two consecutive interior angles measure 100° and 80°. As a check, 100° + 80° = 180°.
Frequently Asked Questions
Are consecutive interior angles always supplementary?
They are supplementary only when the two lines cut by the transversal are parallel. If the lines are not parallel, the two angles still exist on the same side of the transversal between the lines, but they will not add up to 180°. In fact, you can use this property in reverse: if the consecutive interior angles do sum to 180°, that proves the two lines are parallel.
What is the difference between consecutive interior angles and co-interior angles?
There is no difference — they are two names for the same pair of angles. 'Co-interior angles' and 'same-side interior angles' are alternate names commonly used in different textbooks and countries.
Consecutive Interior Angles vs. Alternate Interior Angles
Both types involve interior angles formed by a transversal crossing two lines. Consecutive interior angles are on the same side of the transversal and are supplementary (sum to 180°) when the lines are parallel. Alternate interior angles are on opposite sides of the transversal and are congruent (equal in measure) when the lines are parallel. A common way to remember: 'same side → supplementary' versus 'alternate sides → equal.'
Why It Matters
The consecutive interior angles theorem is one of the key tools for solving angle problems in geometry whenever parallel lines appear. It also works as a test for parallelism: if you can show that two same-side interior angles sum to 180°, you have proven the lines are parallel. This reasoning shows up in proofs, construction problems, and real-world applications like road design and architecture where parallel structures intersect with a cross-beam or path.
Common Mistakes
Mistake: Assuming consecutive interior angles are equal instead of supplementary.
Correction: Consecutive interior angles add up to 180° — they are supplementary, not congruent. It is alternate interior angles that are equal. Remember: same side means supplementary; opposite sides means equal.
Mistake: Applying the supplementary property when the two lines are not parallel.
Correction: The theorem that consecutive interior angles sum to 180° requires the two lines to be parallel. If the lines are not parallel, you cannot assume the angles are supplementary.
Related Terms
Supplementary Angles — Consecutive interior angles are supplementary when lines are parallel
