1. Geometry Basics

Lesson

Let's explore the names and properties when we have two or more angles together.

We can give special names to angle pairs whose measures add to a certain value. When angles have measures adding to $90^\circ$90°, we say they are complementary angles. When angles have measures adding to $180^\circ$180°, we say they are supplementary angles.

The descriptions of complementary and supplementary apply to angle pairs regardless of the location of the angles.

Two angles are supplementary. What do their measures add to?

Their measures add to $\editable{}$ °.

We can describe angle pairs by the location of the two angles relative to one another. Based on properties of these relationships, we can then conclude certain things about the measures of the angles.

Adjacent angles are angles that share a common ray and vertex and do not overlap. The word adjacent means "next to", so these are angles that are next to one another. In the applet below, you can see how the measure of the smaller adjacent angles always equals the measure of the larger angle they make up.

If two adjacent angles form a straight angle, we say that that the angles form a linear pair. Since straight angles measure $180^\circ$180°, linear pairs are always supplementary.

Likewise, two angles forming a right angle are always complementary.

When two lines intersect, they form four distinct, non-overlapping angles. The non-adjacent angles are called vertical angles. It can be proven that all pairs of vertical angles are congruent. They have equal measure.

Let's try applying one of the properties above in an example problem where we solve for a missing value.

Consider the diagram below.

Solve for $x$

`x`.

Let's summarize the different angle pairs we've discussed:

Angle Pair Relationships

Complementary Angles - angles with measures adding to $90^\circ$90°

Supplementary Angles - angles with measures adding to $180^\circ$180°

Adjacent Angles - two angles sharing a common ray and vertex that are not overlapping

Linear Pair - adjacent angles forming a straight angle

And these are the properties of those angle pairs:

Angle Pair Properties

- Adding the measures of two adjacent angles always equals the measure of the larger angle they make up.
- Linear pairs are supplementary.
- Angles forming a right angle are complementary.
- Vertical angles are congruent.

One diagram isn't limited to just one angle pair relationship. We can apply multiple angle pairs at once, like in the practice question below.

Consider the diagram below.

Solve for $x$

`x`.