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7.02 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:

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.

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

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

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.

Worked example

Question 1

Solve for the value of $x$x in the diagram below:

Think: The angle formed is a full revolution, so adding these angles all together will make $360^\circ$360°.
Do: We write the equation:


We then use subtraction to make $x$x the subject:


We then do the subtraction to find $x$x:


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


Practice questions

Question 2

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

Two lines intersect at point X, featuring five points on each. The first line hosts points A, P, X, R, and C in sequential order, while the second line includes points B, Q, X, S, and D, also arranged sequentially.

  1. Use the angle symbol $\angle$ in your answer.

Question 3

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

A right angle, marked by a green box, is divided by a line segment, creating two angles: a top angle and a bottom angle. The top angle, shaded in red, measures $x$x degrees, indicating its unknown measure. The bottom angle, shaded in blue, measures $39^\circ$39°.



Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure.

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