7.01 Angle relationships

Whenever two angles share a defining line, ray, or segment, and do not overlap, we say they are adjacent angles. Here are some examples:

Angles are marked with an arc or a number of arcs.

Whenever two segments, lines, or rays intersect at a point, two pairs of equal angles are formed. Each angle in the pair is on the opposite side of the intersection point, and they are called vertical angles.

Four angles are formed by the intersecting lines, and there are two pairs of equal angles. Each pair are vertical angles.

We can see that equal angles can be denoted by placing an equal number of arcs betwen the rays that form equal angles. Angles marked with different number of arcs are angles of different measurements.

If two angles form a right angle, we say they are complementary. We then know that they add to 90\degree.

If two angles form a straight angle, we say they are supplementary. We then know that they add to 180\degree.

Examples

Example 1

Which of these diagrams shows a pair of adjacent angles?

A

B

C

D

Worked Solution

Create a strategy

We should look for two angles share a defining segment but do not overlap.

Apply the idea

The option that shows the two angles touching along a common segment is option C.

Example 2

Write an angle that is supplementary with \angle CXD in the figure below:

Use the angle symbol \angle in your answer.

Worked Solution

Create a strategy

Supplementary angles are two angles forming a straight angle.

Apply the idea

Any one of the following angles are supplementary to \angle CXD: \angle AXD, \angle PXD, \angle AXS, \angle PXS, \angle CXB, \angle RXB, \angle CXQ, \angle RXQ

Watch question walkthrough

Whenever we know that two (or more) angles form a right angle, a straight angle, or a full revolution, we can write an equation that expresses this relationship.

Let's look at the following examples to find missing angles based on angle relationships.

Examples

Example 3

Solve for the value of x in the diagram below:

Worked Solution

Create a strategy

The angles formed makes a full revolution equivalent to 360\degree.

Apply the idea

\displaystyle x + 116 + 147	\displaystyle =	\displaystyle 360	Equate the sum of angles to 360
\displaystyle x + 263	\displaystyle =	\displaystyle 360	Evaluate addition
\displaystyle x	\displaystyle =	\displaystyle 97	Subtract 263 from both sides

Reflect and check

We never use degrees once we are working with an equation. We are solving for the value of x, and we don't want to double up on using the degree symbol.

Example 4

The angles in the diagram below are complementary. What is the value of x?

Worked Solution

Create a strategy

Complementary angles are two angles forming a right angle equivalent to 90\degree.

Apply the idea

\displaystyle x + 39	\displaystyle =	\displaystyle 90	Equate the sum of the angles to 90
\displaystyle x	\displaystyle =	\displaystyle 51	Subtract 39 from both sides

Watch question walkthrough