Identifying Angle Pair Relationships | Grade 7 Math | LevelKite

In this lesson you name how two angles are related. You already name an angle with three letters, with the vertex in the middle. For example, angle $ABD$ has vertex $B$ and sides $BA$ and $BD$ . You write $m\angle ABD$ for its measure in degrees.

Two angles are adjacent when they share a vertex and a side and do not overlap. Adjacent describes position only. It says nothing about measures. Two angles are supplementary when their measures add to exactly $180^\circ$ . Two adjacent angles form a linear pair when their outer sides point in opposite directions along one line. Together they make a straight angle, so a linear pair is always supplementary.

In the figure below, ray $BD$ starts at point $B$ on line $AC$ . That one ray splits the straight angle at $B$ into angle $ABD$ and angle $DBC$ .

Ray

BD

starts on line

AC

at

B

and splits the straight angle there into angle

ABD

and angle

DBC

.

Worked example

Use the figure with line $AC$ and ray $BD$ . Are angle $ABD$ and angle $DBC$ adjacent? Are they supplementary?

Check the vertex. The middle letter of each name is $B$ , so both angles have vertex $B$ .
List the sides. Angle $ABD$ has sides $BA$ and $BD$ . Angle $DBC$ has sides $BD$ and $BC$ . Side $BD$ appears in both, so the angles share a side.
Check for overlap. Angle $ABD$ opens on one side of ray $BD$ and angle $DBC$ opens on the other side, so their insides do not overlap. A shared vertex, a shared side and no overlap make the angles adjacent.
Look at the outer sides $BA$ and $BC$ . Both lie on line $AC$ and point in opposite directions from $B$ , so the angles form a linear pair. Together they fill a straight angle, which measures $180^\circ$ . Their measures therefore add to $180^\circ$ .
$m\angle ABD + m\angle DBC = 180^\circ$

Answer

Angle $ABD$ and angle $DBC$ are adjacent. Because they form a linear pair, they are also supplementary.

Two angles are complementary when their measures add to exactly $90^\circ$ . Like supplementary, this word is about measures, not position.

You can see a complementary pair in the next figure without using any numbers. The square mark at $B$ shows that rays $BA$ and $BC$ are perpendicular, so angle $ABC$ is a right angle and measures $90^\circ$ . Ray $BD$ lies inside that right angle. Angle $ABD$ and angle $DBC$ share vertex $B$ and side $BD$ and do not overlap, so they are adjacent. Together they fill the right angle exactly, so $m\angle ABD + m\angle DBC = 90^\circ$ . That makes them complementary, whatever the size of each part.

The square mark shows that angle

ABC

is a right angle. Ray

BD

splits it into angle

ABD

and angle

DBC

.

Complementary and supplementary angles do not have to touch. Two angles are separate when they share no vertex. For separate angles, no right-angle mark or line shows their total, so you need their stated measures. Add the measures. A sum of exactly $90^\circ$ means the angles are complementary. A sum of exactly $180^\circ$ means they are supplementary. Any other sum means they are neither.

In the next figure, angle $ABC$ measures $40^\circ$ and angle $DEF$ measures $140^\circ$ . The figure is not drawn to scale, so use these stated values, not the look of the drawing.

Angle

ABC

measures

40^\circ

and angle

DEF

measures

140^\circ

. The two angles share no vertex.

Worked example

Angle $ABC$ measures $40^\circ$ and angle $DEF$ measures $140^\circ$ . Are they adjacent? Are they complementary, supplementary or neither?

Check position. Angle $ABC$ has vertex $B$ and angle $DEF$ has vertex $E$ . The angles share no vertex and no side, so they are not adjacent.
Add the stated measures. Being complementary or supplementary depends only on the total.
$40^\circ + 140^\circ = 180^\circ$
Compare the sum with the two targets. The sum is exactly $180^\circ$ , so the angles are supplementary. The sum is not $90^\circ$ , so the angles are not complementary.

Answer

Angle $ABC$ and angle $DEF$ are not adjacent. They are supplementary because $40^\circ + 140^\circ = 180^\circ$ .

When two lines cross, they make four angles around the crossing point. Two of these angles that sit directly across from each other are vertical angles. They share only the vertex. Each side of one angle points the opposite way from a side of the other angle, along the same line.

Vertical angles are always equal. To see why, use the next figure, where lines $AC$ and $BD$ cross at $E$ . Sides $EB$ and $ED$ form line $BD$ , so angle $AEB$ and angle $AED$ form a linear pair: $m\angle AEB + m\angle AED = 180^\circ$ . Sides $EC$ and $EA$ form line $AC$ , so angle $CED$ and angle $AED$ also form a linear pair: $m\angle CED + m\angle AED = 180^\circ$ . So $m\angle AEB$ and $m\angle CED$ both equal $180^\circ - m\angle AED$ , which means they are equal. The same reasoning shows that angle $AED$ equals angle $BEC$ .

Lines

AC

and

BD

cross at

E

. Angle

AEB

is across from angle

CED

, and angle

AED

is across from angle

BEC

.

Worked example

Lines $AC$ and $BD$ cross at $E$ . Name the relationship for each pair: angle $AEB$ with angle $CED$ , and angle $AEB$ with angle $BEC$ .

Start with angle $AEB$ and angle $CED$ . Both have vertex $E$ . Their sides are $EA$ and $EB$ , and $EC$ and $ED$ . They have no side in common, so they are not adjacent, even though they share a vertex.
Compare the sides. $EA$ and $EC$ point opposite ways along line $AC$ , and $EB$ and $ED$ point opposite ways along line $BD$ . So the angles sit across from each other where the lines cross. They are vertical angles, and vertical angles are equal.
$m\angle AEB = m\angle CED$
Now take angle $AEB$ and angle $BEC$ . They share vertex $E$ and side $EB$ . They lie on opposite sides of $EB$ , so they do not overlap. They are adjacent.
Their outer sides, $EA$ and $EC$ , form line $AC$ . So the two angles form a linear pair and fill a straight angle. That makes them supplementary.
$m\angle AEB + m\angle BEC = 180^\circ$

Answer

Angle $AEB$ and angle $CED$ are vertical angles, so they are equal. Angle $AEB$ and angle $BEC$ are adjacent and form a linear pair, so they are supplementary.

Key concept

Four ways two angles can be related

Adjacent: the angles share a vertex and a side and do not overlap. This describes position, so it does not fix a sum by itself.
Vertical: the angles sit across from each other where two lines cross. Vertical angles are always equal.
Complementary: the measures add to exactly $90^\circ$ . Two adjacent angles that fill a right angle are an example.
Supplementary: the measures add to exactly $180^\circ$ . A linear pair is always supplementary.

Complementary and supplementary angles can be adjacent or separate. One pair of angles can have more than one relationship.

To identify an angle pair, check position first. A shared vertex and a shared side with no overlap means the angles are adjacent. Opposite angles at two crossing lines are vertical, so they are equal. Then check measures. A total of exactly $90^\circ$ means complementary, and a total of exactly $180^\circ$ means supplementary. The total can come from a right-angle mark, a line or stated measures. Next, you will use these relationships to find the measures of complementary and supplementary angles.