7th Grade · 7.G.B.5

Supplementary and complementary angles in 7th grade

Supplementary and complementary angles are pairs of angles that add up to a set number of degrees. In 7th grade, your child learns four angle names: supplementary, complementary, vertical and adjacent. Your child then uses them to write and solve a simple equation for a missing angle.

What the standard says

The official Common Core wording, quoted exactly. The plain version follows.

7.G.B.5 Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure.

What it means, in plain words

An angle is the opening between two lines that meet at a point, measured in degrees. Complementary angles add up to $90^\circ$ , the size of a square corner. Supplementary angles add up to $180^\circ$ , the size of a straight line. The two angles do not have to touch: a $30^\circ$ angle and a $60^\circ$ angle are complementary wherever they sit.

Adjacent angles sit side by side. They share a corner point and one side, and they do not overlap. When two adjacent angles together fill a straight line, they are supplementary. Vertical angles are the angles across from each other where two straight lines cross. They are always equal.

In 7th grade, supplementary and complementary angles show up in diagrams with a letter in place of a number. Your child reads the diagram, picks the angle fact that fits and writes an equation. Many problems need two facts in a row. Solving for the letter is not the end: your child puts the value back in to find the angle itself.

Your child should be able to:

Tell complementary angles ( $90^\circ$ ) from supplementary angles ( $180^\circ$ )
Spot adjacent angles and vertical angles in a diagram
Write an equation for a missing angle and solve it
Use two angle facts in a row, such as a straight line and then vertical angles

Example

Two angles are complementary, and one is twice the size of the other. How big is each angle?

Call the smaller angle $x$ . The larger one is $2x$ .
Complementary angles add up to $90^\circ$ : $x + 2x = 90$ .
$3x = 90$ , so $x = 30$ .
The two angles are $30^\circ$ and $60^\circ$ .

Example

Two straight lines cross. One angle is $70^\circ$ . The angle beside it on the same line is $(2x + 10)^\circ$ . What is $x$ , and what are the four angles?

The two angles sit side by side on a straight line, so they are supplementary: $70 + 2x + 10 = 180$ .
$2x + 80 = 180$ , so $2x = 100$ and $x = 50$ .
The angle beside the $70^\circ$ angle is $2 \times 50 + 10 = 110^\circ$ .
Vertical angles are equal, so $x = 50$ and the four angles are $70^\circ$ , $110^\circ$ , $70^\circ$ and $110^\circ$ .

Where children usually get stuck

Your child writes $x + 40 = 90$ for two angles that together make a straight line.
Complementary and supplementary sound alike, so the child picks one of the two totals without checking the picture. A straight line is $180^\circ$ .
What helps. Share a memory hook: C comes before S in the alphabet, and 90 comes before 180. "Corner" for complementary and "straight" for supplementary also works.
Your child solves the equation, writes $x = 50$ and stops, but the question asked for the size of the angle.
Solving for $x$ feels like the end of the problem. Here $x$ is only part of the angle, which is $(2x + 10)^\circ = 110^\circ$ .
What helps. Ask your child to put the value back into the diagram and check that the angles add up to $90^\circ$ or $180^\circ$ as they should.
Your child marks two side-by-side angles at a crossing as equal, as if they were vertical angles.
The rule "vertical angles are equal" gets remembered as "angles at a crossing are equal". Only the angles straight across from each other are equal. Side-by-side angles at a crossing add up to $180^\circ$ .
What helps. Draw two crossing lines and trace each line with a finger. Vertical angles touch only at the crossing point and share no side.

Try four practice questions

Real 7th grade questions from our question bank, easiest first. Answer them right here, check each one, and see a worked solution for any question.

1Warm-up
Look at the diagram and the two solution attempts. Which student found the correct value of x?
$\displaystyle \begin{array}{l}\text{Alex's work: } (x+20)+55=90\\x+75=90\text{, so }x=15\\ \text{Jordan's work: } x+75=90\\x=15\end{array}$
A
Alex
B
Jordan
C
Both
D
Neither
Show the answer and how to solve it
Both are correct. Alex's (x+20)+55=90 and Jordan's equivalent x+75=90 both model the right-angle relationship and give x=15.
1. 1Look at the angle labeled (x+20)° and the angle directly across the vertex from it. Vertical angles are equal, so the unlabeled upper-left angle is also (x+20)°.
2. 2That upper-left angle and the $55°$ angle together fill the marked right angle, so they must add to $90°$ .
3. 3Write and solve the equation using the right-angle sum. Combining the constants 20 and 55 into 75 first is the same as keeping them separate — either form is valid.
4. 4Alex kept the terms separate as (x+20)+55=90, and Jordan combined 20+55=75 in the first line to write x+75=90. These are the same equation in two forms, and both reach x=15. Both students are correct.
Common mistake. A common mistake is assuming Jordan's work is wrong because the first line looks shorter than Alex's. Combining the two constants 20+55=75 before solving is valid algebra, and both first lines lead to x=15.
The idea underneath. In a multi-step angle problem, you often move a measure across the diagram with one fact first, then use a second fact like a right angle or a straight angle to build the equation.
2On level
Use the angle relationships in the diagram to find x.
x =
Show the answer and how to solve it
Answer. 15
The answer is $15$ because the vertical angle in the upper-right is $55^{\circ}$ , and $55 + (2x + 5) = 90$ .
1. 1The $55^{\circ}$ angle in the lower-left and the unlabeled angle between the horizontal ray and the diagonal in the upper-right are vertical angles. Vertical angles are equal, so that upper-right angle is also $55^{\circ}$ .
2. 2The upper-right corner is marked as a right angle, so its two parts add to $90^{\circ}$ .
3. 3Simplify the left side.
4. 4Subtract $60$ from both sides.
5. 5Divide both sides by $2$ .
Common mistake. A common mistake is subtracting $55$ from $90$ and entering $35$ . That gives the measure of the angle labeled $(2x+5)^{\circ}$ , but you still have to solve $2x + 5 = 35$ to find $x = 15$ .
The idea underneath. In a two-fact angle problem, first turn the hidden angle into a number using an angle relationship, then use the whole right or straight angle to write the equation for x.
3On level
Which equation can you write from the diagram before solving for x?
A
$\displaystyle (x + 7) + {}$
$\displaystyle 25 = {}$
$\displaystyle 90$
B
$\displaystyle x + {}$
$\displaystyle 7 = {}$
$\displaystyle 25$
C
$\displaystyle (x + 7) + {}$
$\displaystyle 25 = {}$
$\displaystyle 180$
D
$\displaystyle 2(x + 7) = {}$
$\displaystyle 90$
Show the answer and how to solve it
The correct equation is $(x + 7) + 25 = 90$ because the upper-right angle equals $(x+7)^{\circ}$ by vertical angles, and that angle plus $25^{\circ}$ makes the marked right angle.
1. 1Look at the marked right angle. The diagonal ray splits that $90^{\circ}$ angle into two smaller angles: one is $25^{\circ}$ , and the other is the upper-right angle.
2. 2The lower-left angle labeled $(x+7)^{\circ}$ and the upper-right angle are vertical angles, so they have the same measure.
3. 3Replace the upper-right angle with $x+7$ . Then write the sum for the right angle.
4. 4So the equation that matches the diagram is $(x + 7) + 25 = 90$ .
Common mistake. Do not set $x+7$ equal to $25$ . The angle labeled $(x+7)°$ is vertical to the unlabeled upper-right angle, not to the angle labeled $25°$ . The upper-right angle and the $25°$ angle are the adjacent parts of the right angle.
The idea underneath. An angle measure can be transferred across the diagram through a vertical-angle match, then used as one part of the marked right angle.
4Stretch
Look at the angle diagram. What is the value of x?
A
25
B
55
C
56
D
75
Show the answer and how to solve it
The answer is $25$ because $(2x+6)+(x+9)=90$ gives $3x+15=90$ , so $x=25$ .
1. 1The square marker shows the whole angle is $90°$ . The two labeled parts together must add to $90°$ .
2. 2Write an equation using both labels.
3. 3Combine like terms on the left side.
4. 4Subtract $15$ from both sides.
5. 5Divide both sides by $3$ to find $x$ .
Common mistake. A common mistake is stopping at $56°$ . That is the measure of the angle labeled $(2x+6)°$ , not the value of $x$ itself.
The idea underneath. When both angle labels contain x, write a single equation using both expressions and collect like terms before isolating x.