Finding Complementary and Supplementary Angle Measures | Grade 7 Math | LevelKite

Some pairs of angles work like partners: their two measures add up to a special total. When you know that total and one of the measures, you can work out the other measure yourself. That is the skill you build in this lesson.

An angle's measure tells how far one side is turned from the other. You count that turn in degrees, so $90^\circ$ means 90 degrees. Two totals matter here: $90^\circ$ is the measure of a right angle, the kind you see at a square corner, and $180^\circ$ is the measure of a straight angle, whose two sides form a straight line. Every angle measure in this lesson is greater than $0^\circ$ .

You may have met the words complementary and supplementary before. Each word names a pair of angles with one exact total.

Two angles are complementary when their measures add to exactly $90^\circ$ . Angles that measure $35^\circ$ and $55^\circ$ are complementary because $35^\circ + 55^\circ = 90^\circ$ . Each angle is the complement of the other.
Two angles are supplementary when their measures add to exactly $180^\circ$ . Angles that measure $110^\circ$ and $70^\circ$ are supplementary because $110^\circ + 70^\circ = 180^\circ$ . Each angle is the supplement of the other.

Only the sum of the two measures decides the name, so you can test any pair by adding.

Add the two stated measures to name the pair

First measure	Second measure	Sum of the measures	Kind of pair
$62^\circ$	$28^\circ$	$62^\circ + 28^\circ = 90^\circ$	Complementary
$45^\circ$	$135^\circ$	$45^\circ + 135^\circ = 180^\circ$	Supplementary
$12.5^\circ$	$77.5^\circ$	$12.5^\circ + 77.5^\circ = 90^\circ$	Complementary
$91^\circ$	$89^\circ$	$91^\circ + 89^\circ = 180^\circ$	Supplementary
$50^\circ$	$60^\circ$	$50^\circ + 60^\circ = 110^\circ$	Neither: the sum is not $90^\circ$ and not $180^\circ$

Now turn the definition around. Suppose you know that two angles are complementary and that one of them measures $40^\circ$ . The definition fixes the total at $90^\circ$ , and the known angle already uses $40^\circ$ of it. The other angle must supply exactly what is left: $90^\circ - 40^\circ = 50^\circ$ . You can confirm it by adding: $40^\circ + 50^\circ = 90^\circ$ .

A supplementary pair works the same way. Only the total changes: it is $180^\circ$ instead of $90^\circ$ .

In both cases, you can write the idea as an equation, a statement that two amounts are equal. The total is $90^\circ$ for a complementary pair and $180^\circ$ for a supplementary pair:

\text{unknown measure} + \text{known measure} = \text{total for that kind of pair}

Key concept

The name of the pair tells you the total

Let $a$ and $b$ stand for the numbers of degrees in two angles, each greater than $0$ .

If the two angles are complementary, then $a + b = 90$ .
If the two angles are supplementary, then $a + b = 180$ .

So when you know the kind of pair and one measure, subtract the known measure from that pair's total to find the other measure. Subtracting the same number from both sides of an equation keeps the two sides equal, which gives $b = 90 - a$ for a complementary pair and $b = 180 - a$ for a supplementary pair. Check by adding: the two measures must total exactly $90^\circ$ for a complementary pair and exactly $180^\circ$ for a supplementary pair.

\text{complementary: } a + b = 90 \qquad \text{supplementary: } a + b = 180

Worked example

Finding a complement

Two angles are complementary. One angle measures $37^\circ$ . What is the measure of the other angle?

Name the unknown before you use it. Let x stand for the number of degrees in the other angle.
Write what complementary means. The two measures of a complementary pair add to exactly 90 degrees, so x plus 37 equals 90.
$x + 37 = 90$
Subtract 37 from both sides to get x by itself. Taking the same number away from two equal amounts leaves two equal amounts, so the equation stays true.
$x + 37 - 37 = 90 - 37$
Do the arithmetic on each side. On the left, 37 minus 37 is 0, so only x remains. On the right, take 30 from 90 to get 60, then take 7 more to get 53.
$x = 53$
Check against the definition. The measures of a complementary pair must add to exactly 90 degrees, and 37 plus 53 does. The answer is also greater than 0, as every angle measure in this lesson must be.
$37 + 53 = 90$

Answer

The other angle measures $53^\circ$ , because $37^\circ + 53^\circ = 90^\circ$ .

Worked example

Finding a supplement

Two angles are supplementary. One angle measures $128^\circ$ . What is the measure of the other angle?

Let x stand for the number of degrees in the other angle. The pair is supplementary, so the two measures add to exactly 180 degrees.
$x + 128 = 180$
Subtract 128 from both sides. Both sides lose the same amount, so they stay equal.
$x + 128 - 128 = 180 - 128$
Do the arithmetic on each side. The left side is x, because 128 minus 128 is 0. On the right, take 100 from 180 to get 80, then take 28 from 80 to get 52.
$x = 52$
Check against the definition: the measures of a supplementary pair must add to exactly 180 degrees, and 128 plus 52 does.
$128 + 52 = 180$

Answer

The other angle measures $52^\circ$ , because $128^\circ + 52^\circ = 180^\circ$ .

Mixing up the two totals. Suppose two angles are complementary and one measures $25^\circ$ . A common slip is to subtract from $180^\circ$ : $180^\circ - 25^\circ = 155^\circ$ . Test that answer by adding: $25^\circ + 155^\circ = 180^\circ$ , not $90^\circ$ . That total belongs to a supplementary pair, so $155^\circ$ is the supplement of the $25^\circ$ angle, not its complement. The word complementary fixes the total at $90^\circ$ , so the correct work is $90^\circ - 25^\circ = 65^\circ$ , and $25^\circ + 65^\circ = 90^\circ$ confirms it. Before you subtract, match the name of the pair to its own total. A memory aid: c for complementary comes before s for supplementary in the alphabet, just as $90$ comes before $180$ .

Worked example

One angle, two different partners

An angle measures $67.5^\circ$ . Find the measure of its complement and the measure of its supplement.

Start with the complement. Let c stand for the number of degrees in the complement. The measures of an angle and its complement add to exactly 90 degrees.
$c + 67.5 = 90$
Subtract 67.5 from both sides, which keeps the two sides equal. Write 90 as 90.0, which has the same value, to line up the decimal points. Then 90.0 minus 67.0 is 23.0, and 23.0 minus 0.5 is 22.5.
$c = 90.0 - 67.5 = 22.5$
Now find the supplement. Let s stand for the number of degrees in the supplement. The measures of an angle and its supplement add to exactly 180 degrees, so the total changes even though the given angle is the same.
$s + 67.5 = 180$
Subtract 67.5 from both sides again, so the two sides stay equal. Then 180.0 minus 67.0 is 113.0, and 113.0 minus 0.5 is 112.5.
$s = 180.0 - 67.5 = 112.5$
Check each answer against its own total: exactly 90 degrees for the complementary pair and exactly 180 degrees for the supplementary pair. Both answers are greater than 0, so both are possible angle measures.
$67.5 + 22.5 = 90 \quad \text{and} \quad 67.5 + 112.5 = 180$

Answer

The complement of the $67.5^\circ$ angle measures $22.5^\circ$ , and its supplement measures $112.5^\circ$ .

Here is the idea to keep: the name of the pair tells you the total. The measures of a complementary pair add to exactly $90^\circ$ , and the measures of a supplementary pair add to exactly $180^\circ$ . When you know the kind of pair and one measure, subtract the known measure from that pair's total. Then add the two measures to check that you get the total back.

Next, you will work with two more kinds of angle pairs, vertical angles and adjacent angles, and you will write and solve equations to find an unknown angle measure.