What Is a Linear Pair of Angles in Geometry?

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.