What Is A Linear Pair? Linear Pair Of Angles Explained

The short version

A linear pair is two angles that sit side by side and share a vertex and a side, with their other two sides forming a straight line.
The two angles in a linear pair always add up to 180°. To find a missing one, subtract from 180: if one angle is 47°, the other is 180 – 47 = 133°.
Every linear pair is supplementary, but supplementary angles don’t have to be a linear pair. They only form one when they’re adjacent.
When two lines cross, they make four angles and four linear pairs.

A linear pair is two adjacent angles whose outer sides form a straight line. Because a straight line measures 180°, the two angles in a linear pair always add up to 180°. Picture a straight road with a driveway coming off it: the angle on the left of the driveway and the angle on the right make a linear pair.

That 180° fact is what homework questions are really testing. Below we cover the linear pair meaning in full, how it differs from supplementary and vertical angles, worked examples from simple subtraction up to algebra, and practice problems with an answer key.

What is a linear pair of angles?

Two angles form a linear pair when all three of these are true:

They share a vertex (the corner point).
They share one side, so they sit right next to each other with no overlap. That’s what “adjacent” means.
Their two outer sides point in exactly opposite directions, making a straight line.

If you drew ray BD coming up from a point B on line AC, you’d get angle ABD and angle DBC. They share vertex B and side BD, and sides BA and BC form the straight line AC. So they’re a linear pair, and angle ABD + angle DBC = 180°.

The rule behind this is often called the linear pair postulate (some textbooks call it the linear pair theorem): if two angles form a linear pair, they’re supplementary. Geometry students use it as a reason in proofs, so it’s worth knowing the name.

Since the two angles have to add to 180°, a linear pair is always one of these:

one acute angle and one obtuse angle, like 60° and 120°, or
two right angles, 90° and 90°.

Two acute angles can’t make a linear pair. Neither can two obtuse angles.

Linear pair vs supplementary angles vs vertical angles

These terms travel together and get confused a lot. Here’s how they compare.

Term	Must be adjacent?	What’s true about the measures	Example
Linear pair	Yes	Add to 180°	Two angles on a straight line with a ray between them
Supplementary angles	No	Add to 180°	A 70° angle in one corner of the page and a 110° angle in another
Adjacent angles	Yes	Any sum	Two angles sharing a side, like 30° and 45°
Vertical angles	No (they’re opposite each other)	Equal	The angles across from each other where two lines cross

The key point: every linear pair is supplementary, but not every pair of supplementary angles is a linear pair. Supplementary is only about the numbers. Linear pair is about the numbers and the position. For a full rundown of the number side, see our post on supplementary angles.

How to find a missing angle in a linear pair

Every one of these problems comes down to one equation: angle 1 + angle 2 = 180°.

Example 1: One angle is 132°. Find the other.
180 – 132 = 48
The other angle is 48°.

Example 2: The angles are (3x + 10)° and (2x – 5)°. Find both.
(3x + 10) + (2x – 5) = 180
5x + 5 = 180
5x = 175
x = 35
First angle: 3 × 35 + 10 = 115°
Second angle: 2 × 35 – 5 = 65°
Check: 115 + 65 = 180. Good.

Example 3: One angle is 4 times the other. Find both.
Call the smaller angle x. The larger is 4x.
x + 4x = 180
5x = 180
x = 36
The angles are 36° and 144°.

Example 4: One angle is 30° more than the other.
x + (x + 30) = 180
2x + 30 = 180
2x = 150
x = 75
The angles are 75° and 105°.

Linear pairs when two lines cross

When two lines intersect, they make four angles. Each angle pairs up with the two angles beside it, so there are four linear pairs in all. The angles across from each other are vertical angles, and they’re equal.

That means one angle tells you all four. Example: Two lines cross, and one angle is 70°.
The two angles next to it: 180 – 70 = 110° each.
The angle across from it: 70° (vertical angles).
All four: 70°, 110°, 70°, 110°. They add to 360°, which is a full turn around the point.

This is also why vertical angles are equal. Both of them form a linear pair with the same neighbor, so both equal 180 minus that neighbor.

Common mistakes

Using 90 instead of 180. Kids mix up linear pairs with complementary angles, which add to 90°. A straight line is 180°.
Calling any supplementary angles a linear pair. If the two angles aren’t touching, they can’t be a linear pair, even if they add to 180°.
Stopping at x. In algebra problems, the question almost always wants the angle measures. If x = 35, the answer is 115° and 65°.
Subtraction slips across zeros. 180 – 47 is 133. Kids who write 143 or 147 need to revisit regrouping, which is a much older skill.
Trusting the picture. Diagrams usually aren’t drawn to scale. If the lines don’t actually form a straight line according to the labels, the angles aren’t a linear pair.

Practice problems

One angle is 25°. Find the other angle in the linear pair.
One of the two angles is a right angle. What’s the other?
One angle is 101.5°. Find its partner.
The two angles are x° and 2x°. Find both.
The angles are (5x)° and (x + 24)°. Find x and both angles.
Two lines cross and one angle is 38°. Find the other three.
A 100° angle and an 80° angle are drawn in separate parts of a diagram. Are they supplementary? Are they a linear pair?
The angles are (2x + 15)° and (3x – 20)°. Find x and both angles.

Answers

180 – 25 = 155°.
90°, since 180 – 90 = 90.
180 – 101.5 = 78.5°.
x + 2x = 180, so 3x = 180 and x = 60. The angles are 60° and 120°.
5x + x + 24 = 180, so 6x = 156 and x = 26. The angles are 5 × 26 = 130° and 26 + 24 = 50°.
142°, 38° and 142°. The neighbors are 180 – 38 = 142, and the opposite angle equals 38.
They’re supplementary because 100 + 80 = 180. They’re not a linear pair because they aren’t adjacent.
2x + 15 + 3x – 20 = 180, so 5x – 5 = 180, 5x = 185 and x = 37. The angles are 2 × 37 + 15 = 89° and 3 × 37 – 20 = 91°. Check: 89 + 91 = 180.

If your kid is stuck on linear pairs

A linear pair is one rule: add to 180. When a kid keeps getting these wrong, the trouble almost always sits in one of the skills underneath that rule.

The first is subtraction with regrouping. A lot of linear pair mistakes are really 180 – 47 mistakes, and that’s a second or third grade skill. The second is knowing what a degree is and why a straight line is 180°. The third, for older kids, is setting up and solving two-step equations like 5x + 5 = 180. If equations are the sticking point, our post on struggling with algebra explains where that usually starts. Our math skills by grade level guide shows when angle measurement is expected. And if angle homework ends in tears more often than not, this guide to math anxiety is worth a read.

Frequently asked questions

What is a linear pair in simple terms?

Two angles that sit next to each other on a straight line. Together they make a straight angle, so they add up to 180°.

Are linear pairs always supplementary?

Yes. Because they form a straight line, their measures always add to 180°, which is the definition of supplementary.

Are supplementary angles always a linear pair?

No. Supplementary angles only need to add to 180°. They form a linear pair only if they’re also adjacent with outer sides that make a straight line.

Can a linear pair be two acute angles?

No. Two acute angles are each less than 90°, so together they’re less than 180°. A linear pair is either one acute and one obtuse angle, or two right angles.

What is the linear pair postulate?

It states that if two angles form a linear pair, then they’re supplementary. Geometry students cite it as a reason in two-column proofs.

What’s the difference between a linear pair and vertical angles?

A linear pair sits side by side and adds to 180°. Vertical angles sit across from each other where two lines cross, and they’re equal.

Find the actual gap

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