Angle Pairs and Their Measures | Geometry | LevelKite

Two angles can be related in two separate ways: by where they sit and by what their measures add up to. You already know how to name an angle and identify its vertex and sides. In this lesson, you will use those parts to identify three kinds of angle pairs by position. Then you will find missing measures for two kinds of angle pairs defined by their sum.

Adjacent angles are two angles in the same plane that share a vertex, share a side, and have no interior points in common. The interior of an angle is the region between its two sides. So adjacent angles sit side by side without overlapping. All three conditions matter: sharing only a vertex is not enough, and neither is sharing a vertex and a side while one angle overlaps the other.

Opposite rays are two rays that start at the same point and go in opposite directions along one line, so together they make up the whole line. A linear pair is two adjacent angles whose noncommon sides are opposite rays. The noncommon sides are the two sides the angles do not share. Every linear pair is adjacent, but an adjacent pair is a linear pair only when its noncommon sides are opposite rays.

Point

B

lies on line

AC

between

A

and

C

. Ray

BD

is the shared side of

\angle ABD

and

\angle DBC

.

Here is how the figure fits these definitions. You are given that $B$ lies on line $AC$ between $A$ and $C$ , and that $D$ is not on line $AC$ . So rays $BA$ and $BC$ are opposite rays.

$\angle ABD$ and $\angle DBC$ share vertex $B$ and side $BD$ . Because $B$ is between $A$ and $C$ , the two angles lie on opposite sides of that shared side, so they have no interior points in common. By the definition of adjacent angles, they are adjacent. Their noncommon sides, rays $BA$ and $BC$ , are opposite rays, so by the definition of a linear pair, they also form a linear pair.

You never used the size of either angle or the look of the drawing. You used only the stated positions.

Vertical angles are two angles whose sides form two pairs of opposite rays. Two lines that intersect at one point form two pairs of vertical angles. Vertical angles share only their vertex, so they are never adjacent. The name comes from vertex, not from the up-and-down direction, so a pair of vertical angles can face any way on the page.

Lines

AC

and

BD

intersect at

E

, and ray

EF

starts at

E

. The example below states the positions you may use.

Worked example

Identify angle pairs by position

In the figure, you are given that lines $AC$ and $BD$ intersect at $E$ , with $E$ between $A$ and $C$ and between $B$ and $D$ . Point $F$ lies in the interior of $\angle BEC$ . For each pair, name every relationship that fits (adjacent, linear pair, vertical) or say none fits: (a) $\angle AEB$ and $\angle AED$ ; (b) $\angle AEB$ and $\angle CED$ ; (c) $\angle BEF$ and $\angle FEC$ ; (d) $\angle AEB$ and $\angle FEC$ .

List the opposite rays first. E is between A and C, so rays EA and EC are opposite rays. E is between B and D, so rays EB and ED are opposite rays. Both facts follow from the definition of opposite rays and the stated positions, not from the drawing.
Pair (a) shares vertex E and side EA. E is between B and D, so B and D lie on opposite sides of line AC, and the angles have no interior points in common. By the definition of adjacent angles, they are adjacent. Their noncommon sides, EB and ED, are opposite rays, so by the definition of a linear pair, they are also a linear pair.
$\angle AEB \text{ and } \angle AED: \text{ adjacent and a linear pair}$
In pair (b), the first angle has sides EA and EB, and the second has sides EC and ED. EA is opposite EC, and EB is opposite ED, so the sides form two pairs of opposite rays. By the definition of vertical angles, the pair is vertical. The angles share no side, so they are not adjacent.
$\angle AEB \text{ and } \angle CED: \text{ vertical}$
Pair (c) shares vertex E and side EF. F is in the interior of angle BEC, so ray EF splits that angle into two parts with no interior points in common. By the definition of adjacent angles, they are adjacent. Their noncommon sides are EB and EC. The ray opposite EB is ED, not EC, so by the definition of a linear pair, they are not a linear pair.
$\angle BEF \text{ and } \angle FEC: \text{ adjacent, not a linear pair}$
Pair (d) shares only vertex E and no side, so it is not adjacent, which also rules out a linear pair. EA is opposite EC, but the ray opposite EB is ED, not EF, so the sides do not form two pairs of opposite rays. By the definition of vertical angles, the pair is not vertical.
$\angle AEB \text{ and } \angle FEC: \text{ none of these}$

Answer

Pair (a) is adjacent and a linear pair, pair (b) is vertical, pair (c) is adjacent but not a linear pair, and pair (d) has none of these relationships.

Now turn from position to measure. Write $m\angle P$ for the measure of $\angle P$ . Two angles are complementary when their measures add up to $90^\circ$ , the measure of a right angle. Two angles are supplementary when their measures add up to $180^\circ$ , the measure of a straight angle, which is an angle whose sides are opposite rays. Each angle in the pair is called a complement or a supplement of the other.

These definitions say nothing about position: the two angles may be adjacent, or they may sit in separate figures. Each definition also works in both directions. If two angles are complementary, their measures add up to $90^\circ$ , and if two angles have measures that add up to $90^\circ$ , they are complementary. The same is true for supplementary angles and $180^\circ$ .

Every measure must be greater than $0^\circ$ and at most $180^\circ$ , so an angle has a complement only if its measure is less than $90^\circ$ , and a supplement only if its measure is less than $180^\circ$ .

Key concept

Position words and measure words

Adjacent, vertical and linear pair describe how two angles are positioned, so you decide them from stated facts about their vertex and sides. Complementary and supplementary describe only what two measures add up to, so you use them to write an equation. Read neither kind of fact from the drawing, and keep every measure greater than $0^\circ$ and at most $180^\circ$ .

\begin{aligned} \text{complementary: } & m\angle P + m\angle Q = 90^\circ \\ \text{supplementary: } & m\angle P + m\angle Q = 180^\circ \end{aligned}

Worked example

Find a complement and a supplement

You are given that $\angle P$ measures $38^\circ$ . Find the measure of a complement of $\angle P$ and the measure of a supplement of $\angle P$ .

Let c be the measure, in degrees, of a complement. The definition of complementary angles, used from the word to the sum, says the two measures add up to 90.
$38 + c = 90$
Subtract 38 from both sides, using the Subtraction Property of Equality.
$38 + c - 38 = 90 - 38 \quad\Rightarrow\quad c = 52$
Let s be the measure, in degrees, of a supplement. The definition of supplementary angles, used from the word to the sum, says the two measures add up to 180.
$38 + s = 180$
Subtract 38 from both sides again, using the same property.
$38 + s - 38 = 180 - 38 \quad\Rightarrow\quad s = 142$
Check each value. 52 and 142 are both greater than 0 and at most 180, and each sum matches its definition, so you keep both values.
$38 + 52 = 90, \qquad 38 + 142 = 180$

Answer

A complement of $\angle P$ measures $52^\circ$ , and a supplement of $\angle P$ measures $142^\circ$ .

The square at

B

marks

\angle ABC

as a right angle. Angle

ABD

measures

(3x + 6)^\circ

, and angle

DBC

measures

(x + 4)^\circ

.

Worked example

Use a right-angle mark

In the figure, the square mark shows that $\angle ABC$ is a right angle. You are given that point $D$ lies in the interior of $\angle ABC$ , that $m\angle ABD = (3x + 6)^\circ$ , and that $m\angle DBC = (x + 4)^\circ$ . Find both measures.

The Angle Addition Postulate says that if D is in the interior of angle ABC, then the measures of angles ABD and DBC add up to the measure of angle ABC. A right angle measures 90° by definition, so that sum is 90°. By the definition of complementary angles, used from the sum to the word, the two angles are complementary.
$m\angle ABD + m\angle DBC = m\angle ABC = 90^\circ$
Substitute the given expressions, using the Substitution Property of Equality.
$(3x + 6) + (x + 4) = 90$
Combine like terms on the left side.
$4x + 10 = 90$
Subtract 10 from both sides (Subtraction Property of Equality), then divide both sides by 4 (Division Property of Equality).
$4x = 80 \quad\Rightarrow\quad x = 20$
Substitute 20 for x in each given expression, again by the Substitution Property of Equality.
$m\angle ABD = 3(20) + 6 = 66^\circ, \qquad m\angle DBC = 20 + 4 = 24^\circ$
Check. 66 and 24 are both greater than 0 and at most 180, and their sum is 90, so you keep x = 20. These two angles happen to be adjacent, but complementary angles do not have to be.
$66 + 24 = 90$

Answer

Angle $ABD$ measures $66^\circ$ , and angle $DBC$ measures $24^\circ$ .

Worked example

Test both totals

You are given $m\angle R = (2x + 20)^\circ$ and $m\angle S = (x - 50)^\circ$ . Can $\angle R$ and $\angle S$ be complementary? Can they be supplementary? Find the measures in each case that works.

First test complementary. If the angles are complementary, the definition says their measures add up to 90. Write that equation, then combine like terms.
$(2x + 20) + (x - 50) = 90 \quad\Rightarrow\quad 3x - 30 = 90$
Add 30 to both sides (Addition Property of Equality), then divide both sides by 3 (Division Property of Equality).
$3x = 120 \quad\Rightarrow\quad x = 40$
Substitute 40 for x and check. The algebra worked, but angle S would measure −10°, which is not greater than 0°. Reject x = 40: the angles cannot be complementary.
$m\angle R = 2(40) + 20 = 100^\circ, \qquad m\angle S = 40 - 50 = -10^\circ$
Now test supplementary. If the angles are supplementary, the definition says their measures add up to 180. Write that equation, then combine like terms.
$(2x + 20) + (x - 50) = 180 \quad\Rightarrow\quad 3x - 30 = 180$
Add 30 to both sides, then divide both sides by 3, using the same properties.
$3x = 210 \quad\Rightarrow\quad x = 70$
Substitute 70 for x and check. Both measures are greater than 0° and at most 180°, and 160 + 20 = 180, so you keep x = 70.
$m\angle R = 2(70) + 20 = 160^\circ, \qquad m\angle S = 70 - 50 = 20^\circ$

Answer

$\angle R$ and $\angle S$ cannot be complementary, but they can be supplementary, with $m\angle R = 160^\circ$ and $m\angle S = 20^\circ$ .

Takeaway: Decide adjacent, vertical and linear pair from the stated positions of the vertex and sides, never from how a drawing looks. To find a complement or a supplement, set the sum of the measures equal to $90^\circ$ or $180^\circ$ , solve, and keep only measures greater than $0^\circ$ and at most $180^\circ$ .

Next, you will solve complementary and supplementary angle word problems and find angle measures using vertical angles and linear pairs.