Pairs of angles

Angles often come in pairs. Two angles may add up to a right angle, or to a straight angle. Two angles may sit side by side, or face each other where two lines cross. Know­ing these pairs lets you find an unknown angle with­out a pro­trac­tor, just by adding and tak­ing away. This les­son names the pairs and shows you how to use them.

Com­ple­men­tary angles#

Two angles whose sum is $90^\circ$ are called com­ple­men­tary angles. Each one is the com­ple­ment of the other.

$30^\circ$ and $60^\circ$ are com­ple­men­tary, because $30 + 60 = 90$.

To find the com­ple­ment of an angle, take it away from $90^\circ$.

$$\text{complement of } 35^\circ = 90^\circ - 35^\circ = 55^\circ$$

Sup­ple­men­tary angles#

Two angles whose sum is $180^\circ$ are called sup­ple­men­tary angles. Each one is the sup­ple­ment of the other.

$110^\circ$ and $70^\circ$ are sup­ple­men­tary, because $110 + 70 = 180$.

To find the sup­ple­ment of an angle, take it away from $180^\circ$.

$$\text{supplement of } 110^\circ = 180^\circ - 110^\circ = 70^\circ$$

A way to remem­ber: c comes before s in the alpha­bet, and 90 comes before 180. Com­ple­men­tary goes with 90, sup­ple­men­tary with 180.

Exam­ple 1#

Two com­ple­men­tary angles are in the ratio $2 : 3$. Find them.

Let the angles be $2k$ and $3k$. They add up to $90^\circ$:

$$2k + 3k = 90 \quad \text{so} \quad 5k = 90 \quad \text{and} \quad k = 18$$

The angles are $36^\circ$ and $54^\circ$.

Exam­ple 2#

An angle is equal to its own com­ple­ment. Find it.

Let the angle be $x$. Its com­ple­ment is $90 - x$.

$$x = 90 - x \quad \text{so} \quad 2x = 90 \quad \text{and} \quad x = 45$$

The angle is $45^\circ$.

Exam­ple 3#

The sup­ple­ment of an angle is 4 times its com­ple­ment. Find the angle.

Let the angle be $x$. Its sup­ple­ment is $180 - x$ and its com­ple­ment is $90 - x$.

$$\begin{aligned} 180 - x &= 4(90 - x) \\ 180 - x &= 360 - 4x \\ 4x - x &= 360 - 180 \\ 3x &= 180 \\ x &= 60 \end{aligned}$$

The angle is $60^\circ$. Check: its sup­ple­ment is $120^\circ$, its com­ple­ment is $30^\circ$, and $120 = 4 \times 30$.

Adja­cent angles#

Two angles are adja­cent if

1. they have the same ver­tex,
2. they have one arm in com­mon, and
3. they do not over­lap: their other arms are on oppo­site sides of the com­mon arm.

Pic­ture rays $\overrightarrow{OA}$, $\overrightarrow{OB}$ and $\overrightarrow{OC}$ from one point $O$, with $\overrightarrow{OB}$ in between the other two. Then $\angle AOB$ and $\angle BOC$ are adja­cent. Their com­mon arm is $\overrightarrow{OB}$, and together they make $\angle AOC$:

$$\angle AOB + \angle BOC = \angle AOC$$

Lin­ear pair#

If two adja­cent angles have their other two arms in one straight line, they are called a lin­ear pair.

The two arms form a straight angle, so the angles of a lin­ear pair add up to $180^\circ$. A lin­ear pair is always sup­ple­men­tary.

Think of a ray stand­ing on a straight line. It splits the straight angle into two parts, and the two parts must add up to $180^\circ$.

Exam­ple 4#

Two angles form a lin­ear pair. They are $(3x + 10)^\circ$ and $(2x + 20)^\circ$. Find them.

$$\begin{aligned} (3x + 10) + (2x + 20) &= 180 \\ 5x + 30 &= 180 \\ 5x &= 150 \\ x &= 30 \end{aligned}$$

The angles are $3 \times 30 + 10 = 100$ degrees and $2 \times 30 + 20 = 80$ degrees. Check: $100 + 80 = 180$.

Ver­ti­cally oppo­site angles#

When two lines cross, they make four angles at the point where they meet. The angles that face each other across the point are called ver­ti­cally oppo­site angles. There are two such pairs.

Ver­ti­cally oppo­site angles are equal.

Why. Call the four angles, going round in order, $\angle 1$, $\angle 2$, $\angle 3$ and $\angle 4$. So $\angle 1$ and $\angle 3$ face each other, and so do $\angle 2$ and $\angle 4$.

$\angle 1$ and $\angle 2$ lie along one of the lines, so they form a lin­ear pair: $\angle 1 + \angle 2 = 180^\circ$.

$\angle 2$ and $\angle 3$ lie along the other line, so they form a lin­ear pair too: $\angle 2 + \angle 3 = 180^\circ$.

Both sums are $180^\circ$ and both con­tain $\angle 2$. So $\angle 1 = \angle 3$. In the same way, $\angle 2 = \angle 4$.

Exam­ple 5#

Two lines cross, and one of the angles is $40^\circ$. Find the other three.

The angle oppo­site it is also $40^\circ$. Each angle next to it forms a lin­ear pair with it, so each is $180^\circ - 40^\circ = 140^\circ$.

Going round, the four angles are $40^\circ$, $140^\circ$, $40^\circ$ and $140^\circ$. They add up to $360^\circ$.

Angles at a point#

All the angles around a point, with no gaps and no over­laps, add up to one com­plete turn, $360^\circ$.

Exam­ple 6#

Three angles meet at a point. Two of them are $100^\circ$ and $120^\circ$. Find the third.

$$360^\circ - 100^\circ - 120^\circ = 140^\circ$$

Sum­mary#

Prac­tice#

1. Find the com­ple­ment of $28^\circ$.
2. Find the sup­ple­ment of $135^\circ$.
3. Are $65^\circ$ and $25^\circ$ com­ple­men­tary? Are $65^\circ$ and $115^\circ$ sup­ple­men­tary?
4. Two sup­ple­men­tary angles are in the ratio $4 : 5$. Find them.
5. An angle is equal to its sup­ple­ment. Find it.
6. An angle is $20^\circ$ more than its com­ple­ment. Find the angle.
7. Two angles of a lin­ear pair are $(2x + 15)^\circ$ and $(x + 30)^\circ$. Find $x$ and the two angles.
8. Two lines cross and one angle is $72^\circ$. Find the other three angles.
9. Four angles meet at a point. Three of them are $85^\circ$, $95^\circ$ and $110^\circ$. Find the fourth.
10. The com­ple­ment of an angle is one third of its sup­ple­ment. Find the angle.

Answers#