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 $$90°.
If two angles form a straight angle, we say they are supplementary. We then know that they add to $$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 $$x in the diagram below:
Do: We write the equation:
$$x+147+116=360
We then use subtraction to make $$x the subject:
$$x=360−147−116
We then do the subtraction to find $$x:
$$x=97
Reflect: We never use degrees once we are working with an equation. We are solving for the value of $$x, and we don't want to double up on using the degree symbol!
Practice questions
Question 2
Enter an angle that is supplementary with $$∠CXD in the figure below:
Two lines, each with five points labeled, intersect at a common point $$X. From top-left to bottom-right, line $$AC has points $$A, $$P, $$X, $$R and $$C, respectively. From bottom-left to top-right, line $$BD has points $$B, $$Q, $$X, $$S and $$D, consecutively. Angle $$∠CXD is located on the right of the intersection at point $$X.
Angle $$∠CXD is adjacent to angle $$∠BXC and lie on the same line $$BD. Angle $$∠CXD is also adjacent to angle $$∠BXR and lie on the same line $$BD. Angle $$∠CXD is also adjacent to angle $$∠QXC and lie on the same line $$BD. Angle $$∠CXD is also adjacent to angle $$∠QXR and lie on the same line $$BD. Angle $$∠CXD is also adjacent to angle $$∠AXD and lie on the same line $$AC. Angle $$∠CXD is also adjacent to angle $$∠AXS and lie on the same line $$AC. Angle $$∠CXD is also adjacent to angle $$∠PXD and lie on the same line $$AC. Angle $$∠CXD is also adjacent to angle $$∠PXS and lie on the same line $$AC.
Use the angle symbol $$∠ in your answer.
Question 3
The angles in the diagram below are complementary. What is the value of $$x?
A right angle, marked by a green box, is divided by a line segment, creating two angles: a top angle and a bottom angle. The top angle, shaded in red, measures $$x degrees, indicating its unknown measure. The bottom angle, shaded in blue, measures $$39°.