Angle Pairs: Complementary, Vertical & More

Angle pairs are two angles with a special relationship. Complementary angles add up to $90^\circ$ . Supplementary angles add up to $180^\circ$ . A linear pair is two adjacent angles that form a straight line, and vertical angles are the opposite angles made by two crossing lines. Vertical angles are always congruent.

Before this lesson: You should know Measuring and Classifying Angles and how to solve multi-step equations.

The Four Main Angle Pairs

Some pairs of angles show up again and again in geometry. Once you spot the pair, you know an equation to write. That equation lets you find a missing angle.

Pair	What it means	Equation
Complementary	Two angles whose measures add up to $90^\circ$	$m\angle 1 + m\angle 2 = 90^\circ$
Supplementary	Two angles whose measures add up to $180^\circ$	$m\angle 1 + m\angle 2 = 180^\circ$
Linear pair	Two adjacent angles whose outer sides form a straight line	$m\angle 1 + m\angle 2 = 180^\circ$
Vertical angles	Opposite angles formed by two crossing lines	$m\angle 1 = m\angle 3$

Adjacent angles share a vertex and a side, and they don’t overlap. Complementary and supplementary angles can be adjacent, but they don’t have to be. A linear pair is always adjacent.

Two diagrams side by side. Left: a right angle split by a ray into a 35 degree angle and a 55 degree angle, labeled complementary, 35 + 55 = 90. Right: a straight line split by a ray into a 120 degree angle and a 60 degree angle, labeled linear pair, supplementary, 120 + 60 = 180. — Complementary angles make a right angle. A linear pair makes a straight line, so its angles are supplementary.

Vertical Angles Theorem

When two lines cross, they make four angles. The angles across from each other are vertical angles. The vertical angles theorem says vertical angles are congruent. For a full two-column proof, see proving angle relationships.

Here is why. In the picture below, $\angle 1$ and $\angle 2$ form a linear pair, so they add up to $180^\circ$ . So do $\angle 2$ and $\angle 3$ . Both $\angle 1$ and $\angle 3$ are what is left after taking $m\angle 2$ away from $180^\circ$ . So they are equal.

Two lines crossing at a point, making four angles numbered 1 to 4 around the point. Angle 1 and angle 3 are across from each other and are both labeled 65 degrees with blue arcs. Angle 2 and angle 4 are across from each other and are both labeled 115 degrees with orange dashed arcs. — Vertical angles match: ∠1 = ∠3 = 65° and ∠2 = ∠4 = 115°. Side-by-side angles form linear pairs that add to 180°.

Steps to solve an angle pair problem

Name the pair: complementary, supplementary, linear pair or vertical.
Write the equation: sum = $90^\circ$ , sum = $180^\circ$ , or the two angles are equal.
Solve the equation.
Put $x$ back in to find each angle.
Check the sum or the match.

Worked Examples

Example 1: Complement and supplement

Find the complement and the supplement of a $34^\circ$ angle.

Step 1: The complement adds to $90^\circ$ . Subtract.

90 - 34 = 56

Step 2: The supplement adds to $180^\circ$ . Subtract.

180 - 34 = 146

The complement is $56^\circ$ and the supplement is $146^\circ$ .

Example 2: Angles made by two crossing lines

Two lines cross. One of the angles is $65^\circ$ . Find the other three angles.

Step 1: The angle across from it is a vertical angle, so it is congruent.

m\angle 3 = 65^\circ

Step 2: The angle next to it forms a linear pair with it. Subtract from $180^\circ$ .

m\angle 2 = 180 - 65 = 115^\circ

Step 3: The last angle is vertical to $\angle 2$ .

m\angle 4 = 115^\circ

Step 4: Check: all four angles go all the way around the point.

65 + 115 + 65 + 115 = 360

The angles are $65^\circ$ , $115^\circ$ , $65^\circ$ and $115^\circ$ , as in the picture above.

Example 3: Complementary angles with x

$\angle A$ and $\angle B$ are complementary. $m\angle A = (2x + 8)^\circ$ and $m\angle B = (3x - 3)^\circ$ . Find both angles.

Step 1: Complementary angles add up to $90^\circ$ .

(2x + 8) + (3x - 3) = 90

Step 2: Combine like terms (see combining like terms).

5x + 5 = 90

Step 3: Subtract 5 from both sides.

5x = 85

Step 4: Divide both sides by 5.

x = 17

Step 5: Find $m\angle A$ .

2(17) + 8 = 34 + 8 = 42

Step 6: Find $m\angle B$ .

3(17) - 3 = 51 - 3 = 48

Step 7: Check the sum.

42 + 48 = 90

So $m\angle A = 42^\circ$ and $m\angle B = 48^\circ$ .

Example 4: Vertical angles with x

Two lines cross. Two vertical angles measure $(5x - 12)^\circ$ and $(3x + 20)^\circ$ . Find $x$ , the vertical angles, and the angle next to them.

Step 1: Vertical angles are congruent. Set them equal.

5x - 12 = 3x + 20

Step 2: Subtract $3x$ from both sides (see equations with variables on both sides).

2x - 12 = 20

Step 3: Add 12 to both sides.

2x = 32

Step 4: Divide both sides by 2.

x = 16

Step 5: Find one vertical angle.

5(16) - 12 = 80 - 12 = 68

Step 6: Check with the other.

3(16) + 20 = 48 + 20 = 68

Step 7: The angle next to them forms a linear pair. Subtract from $180^\circ$ .

180 - 68 = 112

So $x = 16$ , the vertical angles are $68^\circ$ , and the angle next to them is $112^\circ$ .

Example 5: Word problem

A ladder leans against a wall. The angle the ladder makes with the ground and the angle it makes with the wall are complementary. The ground angle is 4 times the wall angle. Find both angles.

Step 1: Let the wall angle be $x$ degrees. The ground angle is $4x$ .

Step 2: Complementary angles add up to $90^\circ$ .

x + 4x = 90

Step 3: Combine like terms.

5x = 90

Step 4: Divide both sides by 5.

x = 18

Step 5: Find the ground angle.

4(18) = 72

Step 6: Check the sum.

18 + 72 = 90

The wall angle is $18^\circ$ and the ground angle is $72^\circ$ .

Common Mistakes

Mistake	Example of the error	Fix
Mixing up complementary and supplementary	Saying the complement of $34^\circ$ is $146^\circ$	Complementary is $90^\circ$ , supplementary is $180^\circ$ . Memory trick: C comes before S, and 90 comes before 180.
Adding vertical angles to 180	Writing $(5x - 12) + (3x + 20) = 180$ for vertical angles	Vertical angles are equal. Set them equal to each other.
Calling any two angles next to each other a linear pair	Two adjacent angles of $40^\circ$ and $70^\circ$	A linear pair must form a straight line, so the angles add to $180^\circ$ .
Stopping at $x$	Answering $x = 16$ in Example 4	Put $x$ back in: the angles are $68^\circ$ .
Thinking vertical means up and down	Looking for angles that stand upright	Vertical angles share a vertex and sit across from each other, in any direction.

More Angle Pair Facts

Two acute angles can be complementary, but never supplementary. Each is less than $90^\circ$ , so the sum is less than $180^\circ$ .
If two angles are supplementary and congruent, each is $90^\circ$ . So when two lines cross and one angle is right, all four angles are right. The lines are perpendicular, like in parallel and perpendicular lines.
The four angles around a point add up to $360^\circ$ .
Some problems say “an angle is 20° more than 3 times its supplement.” Write the supplement as $180 - x$ and solve.

Practice Problems

Find the complement of a $27^\circ$ angle.
Find the supplement of a $104^\circ$ angle.
An angle is twice its complement. Find the angle.
$\angle 1$ and $\angle 2$ form a linear pair. $m\angle 1 = (6x + 4)^\circ$ and $m\angle 2 = (4x - 4)^\circ$ . Find both angles.
Two vertical angles measure $(7x - 9)^\circ$ and $(4x + 21)^\circ$ . Find their measure.
Two lines cross, and one angle is $128^\circ$ . Find the other three angles.
$\angle P$ and $\angle Q$ are complementary. $m\angle P = (x + 15)^\circ$ and $m\angle Q = 2x^\circ$ . Find both angles.
An angle is $20^\circ$ more than 3 times its supplement. Find the angle and its supplement.
Can two acute angles be supplementary? Explain.
The blades of open scissors cross like two lines. The angle between the blades is $38^\circ$ . Find the other three angles where they cross.

Show answers

Subtract from 90: $90 - 27 = 63$ . The complement is $63^\circ$ .
Subtract from 180: $180 - 104 = 76$ . The supplement is $76^\circ$ .
Let the angle be x. Its complement is 90 − x. Write the equation: $x = 2(90 - x)$ . Distribute the 2 (see distributive property): $x = 180 - 2x$ . Add 2x to both sides: $3x = 180$ . Divide both sides by 3: $x = 60$ . The angle is $60^\circ$ . Check: its complement is $30^\circ$ , and $2 \cdot 30 = 60$ .
A linear pair adds to 180: $(6x + 4) + (4x - 4) = 180$ . Combine like terms: $10x = 180$ . Divide both sides by 10: $x = 18$ . Find the first angle: $6(18) + 4 = 112$ . Find the second angle: $4(18) - 4 = 68$ . Check: $112 + 68 = 180$ . So $m\angle 1 = 112^\circ$ and $m\angle 2 = 68^\circ$ .
Vertical angles are equal: $7x - 9 = 4x + 21$ . Subtract 4x from both sides: $3x - 9 = 21$ . Add 9 to both sides: $3x = 30$ . Divide both sides by 3: $x = 10$ . Find the angle: $7(10) - 9 = 61$ . Check: $4(10) + 21 = 61$ . Each angle is $61^\circ$ .
The vertical angle matches: $128$ . The angles next to it form linear pairs: $180 - 128 = 52$ . The other angles are $52^\circ$ , $128^\circ$ and $52^\circ$ .
Complementary angles add to 90: $(x + 15) + 2x = 90$ . Combine like terms: $3x + 15 = 90$ . Subtract 15 from both sides: $3x = 75$ . Divide both sides by 3: $x = 25$ . Find each angle: $25 + 15 = 40 \text{ and } 2(25) = 50$ . Check: $40 + 50 = 90$ . So $m\angle P = 40^\circ$ and $m\angle Q = 50^\circ$ .
Let the angle be x. Its supplement is 180 − x. Write the equation: $x = 3(180 - x) + 20$ . Distribute the 3: $x = 540 - 3x + 20$ . Combine the numbers: $x = 560 - 3x$ . Add 3x to both sides: $4x = 560$ . Divide both sides by 4: $x = 140$ . Find the supplement: $180 - 140 = 40$ . The angle is $140^\circ$ and its supplement is $40^\circ$ . Check: $3 \cdot 40 + 20 = 140$ .
No. An acute angle is less than 90°. Two of them add up to less than 180°, so they can never be supplementary.
The vertical angle matches: $38$ . The angles next to it form linear pairs: $180 - 38 = 142$ . The other angles are $142^\circ$ , $38^\circ$ and $142^\circ$ .

FAQ

What is the difference between complementary and supplementary angles?

Complementary angles add up to $90^\circ$ . Supplementary angles add up to $180^\circ$ .

Are vertical angles always congruent?

Yes. That is the vertical angles theorem. Each one forms a linear pair with the same angle between them, so they must be equal.

Is a linear pair always supplementary?

Yes. The two angles form a straight line, which is $180^\circ$ . But supplementary angles don’t have to be a linear pair. They can be far apart.

Do complementary angles have to be next to each other?

No. Any two angles that add up to $90^\circ$ are complementary, even in different places.

Summary

Complementary angles add to $90^\circ$ . Supplementary angles add to $180^\circ$ .
A linear pair is two adjacent angles that form a straight line, so they are supplementary.
Vertical angles are the opposite angles made by two crossing lines. They are congruent.
Name the pair, write its equation, solve for $x$ , then find each angle.
The four angles around a point add up to $360^\circ$ .

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