What Is a Linear Pair of Angles in Geometry?

In geometry, a straight angle has a value of 180° or π radians. However, it doesn't appear to be an angle at all because the lines or opposite rays that meet at the common vertex form a straight line. In this way, a linear pair is very closely related to a straight angle.

A linear pair can also be thought of as straight angle which has a single line segment added to its common vertex. In the process, a straight angle is effectively split into two angles.

If two angles add up to 180°, we call them supplementary angles. At first, this naming seems redundant because linear pairs add up the same value.

However, there is a subtle difference between the two: Supplementary angles do not necessarily have to be adjacent to one another. They can have non-common arms, or not be attached to each other at all.

Linear pairs of angles are always adjacent to one another, and always sharing a common arm. In this way, a linear pair can alternatively be defined as a pair of supplementary adjacent angles. Depending on the author, your geometry textbook may also use supplementary angles as a synonym for a linear pair.

Since we know that a linear pair is made up of exactly two adjacent angles, and we know that these angles add up to a specific value, we can quickly find the value of a second adjacent angle as long as one of the values is already known.

For example, let's we have a linear pair of angles and one of them has a given value of 73°:

As all linear pairs add up to 180°, any geometry problem involving a linear pair of angles can be solved just by subtracting the known angle value from our total of 180 and getting a final value for the linear pair angle where our lines intersect.

In cases where one part of our linear pair is a right angle equal to 90°, solving this problem for the other angle will be even easier. Since 90 is exactly half of 180, our linear pair will be congruent angles with the exact same angle measurement of 90°.

Adjacent angles can be thought of as any two or more angles which meet up at a single point. In the case of a linear pair, these angles add up to 180°, but they can also equal 90°, 360° or some arbitrary angle measurement.

In this way, all linear pairs are adjacent angles, but not all adjacent angles form a linear pair.

Let's go over some different types of adjacent angles, how they differ from a linear pair, and how they can help us find unknown angle values.

Complementary Adjacent Angles

Similarly to how supplementary angles and linear pairs make a 180° value, complementary adjacent angles are two angles that share a common vertex and a common arm, adding up to a 90° value.

You may also think of complementary angles as a single right angle split into two angles by a dividing line segment. Instead of forming a straight line, the non-common arms form a square edge.

Complementary angles can be solved in much the same way that a linear pair of angles can. Instead of subtracting our known angle value from 180° to find the other angle value, we must subtract it from 90°.

If the complementary angles are split perfectly down the middle, two congruent angles form with values of 45° each.

Vertical Angles

Vertical angles are created by crossing two lines or rays over each other. The result is four adjacent angles that all add up to a value of 360°, or 2π radians.

Vertical angles can also be thought of as two linear pairs which are adjacent and congruent to each other. Knowing these rules, we can solve three unknown angle values using only one known angle value.

Solving Vertical Angles

There are a few ways we can go about solving vertical angles. In this example, angles AB and BC form a linear pair, but AB and AD could also be considered to form linear pairs. Since AB and CD are opposite angles, we also know them to be congruent to each other. The same goes for BC compared to AD.

Let's once again assume that angle AB is 73°. Consequently, angle CD must have an equal value of 73°. By subtracting 73° from linear pair value of 180, we can now find the value of both BC and AD.