Identify Angle Relationships in a Figure | Grade 8 Math | LevelKite

When two angles appear in the same figure, they are often related in a way that has a name. In this lesson you will identify four of these relationships: adjacent, vertical, complementary and supplementary angles.

To say exactly which angle you mean, name it with three points, with the vertex in the middle. In $\angle ABC$ , the vertex is $B$ , and the sides are rays $BA$ and $BC$ , which start at $B$ and pass through $A$ and $C$ . Angles are measured in degrees, and $m\angle ABC$ means the measure of $\angle ABC$ .

Two of these relationships depend on where the angles sit.

Adjacent angles share a vertex and a side, and they do not overlap. Adjacent means next to: the shared side separates the two angles, so they touch only along it.
Vertical angles are the two angles directly across from each other where two lines cross. They share a vertex but no side, and their four sides make up the two lines. Here vertical refers to the shared vertex, not to up and down, so vertical angles can face any direction.

The other two relationships depend only on measures.

Complementary angles are two angles whose measures add to exactly $90^\circ$ , the measure of a right angle. In a figure, a small square drawn at a vertex marks a right angle.
Supplementary angles are two angles whose measures add to exactly $180^\circ$ , the measure of a straight angle. A straight angle is formed by two rays that start at the same point and go in opposite directions, so together they make a straight line.

Only the sum matters, so these pairs may share a side or sit apart. For example, if $m\angle PQR = 25^\circ$ and $m\angle STU = 65^\circ$ , then $\angle PQR$ and $\angle STU$ are complementary because $25^\circ + 65^\circ = 90^\circ$ , even though they have different vertices and are not adjacent.

Key concept

Point to the evidence, not the look

To name a pair, point to evidence in the figure or in the stated measures:

Adjacent: a shared vertex and a shared side, with no overlap.
Vertical: two crossing lines, with the angles directly across from each other.
Complementary: two adjacent angles that together make up a marked right angle, or two angles whose stated measures add to $90^\circ$ .
Supplementary: two adjacent angles whose outer sides (the sides they do not share) form a straight line, or two angles whose stated measures add to $180^\circ$ .

One pair can have more than one name. Figures are not drawn to scale, so how big an angle looks is never evidence.

Line

EH

passes through

F

, and ray

FG

starts at

F

, so

\angle EFG

and

\angle GFH

both have vertex

F

.

Worked example

A ray from a point on a line

In the figure above, points $E$ , $F$ and $H$ lie on line $EH$ , and ray $FG$ starts at $F$ . Identify the relationships between $\angle EFG$ and $\angle GFH$ , and give a reason for each.

Look for a shared vertex and a shared side. Both angles have vertex F, and ray FG is a side of each angle.
Check for overlap. Ray FG separates the two angles, so they touch only along that ray. A shared vertex, a shared side and no overlap make the angles adjacent.
Look at the outer sides, rays FE and FH. They start at F and go in opposite directions along line EH, so together the two angles make up the straight angle EFH, which measures 180°. Their measures therefore add to 180°, so the angles are supplementary.
$m\angle EFG + m\angle GFH = m\angle EFH = 180^\circ$

Answer

$\angle EFG$ and $\angle GFH$ are adjacent, because they share vertex $F$ and side $FG$ without overlapping. They are also supplementary, because together they make up a straight angle of $180^\circ$ .

The square mark at

B

shows that

\angle ABD

is a right angle, and ray

BC

lies inside that right angle.

Worked example

A ray inside a right angle

In the figure above, rays $BA$ , $BC$ and $BD$ start at $B$ , and the square mark shows that $\angle ABD$ is a right angle. Decide whether each pair is adjacent and whether it is complementary: (a) $\angle ABC$ and $\angle CBD$ ; (b) $\angle ABC$ and $\angle ABD$ .

(a) Both angles have vertex B and side BC. Ray BC separates them, so they touch only along that ray and do not overlap. The pair is adjacent.
Ray BC lies inside the marked right angle ABD, so angles ABC and CBD together make up angle ABD. The square mark says angle ABD measures exactly 90°, so the two measures add to 90° and the pair is complementary.
$m\angle ABC + m\angle CBD = m\angle ABD = 90^\circ$
(b) Both angles have vertex B and side BA, but because ray BC is inside the right angle, angle ABC lies inside angle ABD. The angles overlap, so the pair is not adjacent.
Angle ABD alone measures 90°, and angle ABC measures more than 0°, so the two measures add to more than 90°. The pair is not complementary.
$m\angle ABC + m\angle ABD = m\angle ABC + 90^\circ > 90^\circ$

Answer

$\angle ABC$ and $\angle CBD$ are adjacent and complementary. $\angle ABC$ and $\angle ABD$ are neither: they overlap, and their measures add to more than $90^\circ$ .

Lines

AC

and

BD

cross at point

E

, making four angles around

E

.

Worked example

Two crossing lines

In the figure above, lines $AC$ and $BD$ cross at point $E$ . Identify how each pair is related: (a) $\angle AEB$ and $\angle CED$ ; (b) $\angle AEB$ and $\angle BEC$ .

(a) Both angles have vertex E, but angle AEB has sides EA and EB, while angle CED has sides EC and ED. They share no side, so they are not adjacent.
Rays EA and EC point in opposite directions along line AC, and rays EB and ED point in opposite directions along line BD. So each side of angle CED points opposite a side of angle AEB, and the two angles sit directly across from each other where the lines cross. They are vertical angles. Likewise, angles BEC and DEA form the other vertical pair.
(b) The angles share vertex E and side EB, which separates them, so they are adjacent. Their outer sides, rays EA and EC, form line AC, so together the angles make up the straight angle AEC and are supplementary.
$m\angle AEB + m\angle BEC = m\angle AEC = 180^\circ$

Answer

$\angle AEB$ and $\angle CED$ are vertical angles, because they sit across from each other where lines $AC$ and $BD$ cross. $\angle AEB$ and $\angle BEC$ are adjacent and supplementary, because they share side $EB$ and their outer sides form line $AC$ .

Angle

ABC

measures

41^\circ

and angle

CBD

measures

47^\circ

; no square mark appears at

B

.

Name an angle relationship only when you can point to its evidence: a shared vertex and a shared side with no overlap, two crossing lines, a square mark, a straight line, or stated measures. How big an angle looks is never evidence. Next, you will use these relationships to find unknown angle measures, write and solve equations, and find and correct errors.