Geometry and Measurement · Shape and Space · Lesson 2

Complementary and Supplementary Angles

Some angles come in pairs that fit together perfectly: two angles that fill a corner, or two that fill a straight line. Carpenters, tailors and builders in every Ghanaian town use these pairs to cut wood and cloth neatly. In this lesson you will learn the two special pairs, complementary and supplementary angles, and use them to find missing angles.

NaCCA B7.3.1.1.2 ⏱ about 45 min

By the end you will be able to

✔State that complementary angles add up to 90° and supplementary angles add up to 180°.
✔Find the complement or supplement of a given angle.
✔Form and solve simple equations to find unknown angles in complementary and supplementary pairs.
✔Decide whether two given angles are complementary, supplementary or neither.

Key words

Complementary angles: Two angles whose sum is 90°. Each is the complement of the other.
Supplementary angles: Two angles whose sum is 180°. Each is the supplement of the other.
Complement: What must be added to an angle to make 90°; the complement of x is 90° − x.
Supplement: What must be added to an angle to make 180°; the supplement of x is 180° − x.
Unknown angle: An angle whose size is not given and is shown by a letter such as x or by a question mark.

Curriculum reference

Content standard: B7.3.1.1 Demonstrate understanding of angles including adjacent, vertically opposite, complementary, supplementary and use them to solve problems.

Indicator: Apply the fact that (i) complementary angles are two angles that have a sum of 90°, and (ii) supplementary angles are two angles that have a sum of 180° to solve problems.

Starter

The carpenter's frame at Suame Magazine

A carpenter at Suame Magazine in Kumasi is making a door frame. He cuts a 90° corner and then, to make a decorative bracket, he cuts the corner into two pieces of wood. One piece makes an angle of 35° with the wall. 'How big is the other?' asks his apprentice. Without any measuring, the carpenter answers: '55°, because the two together must fill the corner of 90°.' This quick thinking is exactly what complementary angles are about.

1. Complementary angles: pairs that make a right angle

When two angles are put side by side, we can ask: what is their total? If the total is exactly 90°, a right angle, we say the angles are complementary. Each angle is called the complement of the other. For example 30° and 60° are complementary because 30° + 60° = 90°.

To find the complement of an angle, subtract it from 90°. The complement of 38° is 90° − 38° = 52°. The complement of 72° is 18°. Notice something important: both angles in a complementary pair must be acute, since each is less than 90°.

▶ The two angles fit together to make a right angle. If one is 35°, the other must be 90° − 35° = 55°.

Complements of some angles
Angle	Its complement
10°	80°
25°	65°
45°	45°
62°	28°
89°	1°

2. Supplementary angles: pairs that make a straight line

Two angles whose sum is 180°, a straight angle, are called supplementary. Each angle is the supplement of the other. 120° and 60° are supplementary, since 120° + 60° = 180°.

To find the supplement of an angle, subtract it from 180°. The supplement of 115° is 180° − 115° = 65°. A supplementary pair can be two right angles (90° + 90°), or one acute and one obtuse angle (like 65° and 115°). It can never be two acute angles, because their total would be less than 180°, and never two obtuse angles, because their total would be more than 180°.

▶ The two angles on a straight line always add up to 180°. Notice the 55° and 125° pair.

Supplements of some angles
Angle	Its supplement
30°	150°
90°	90°
115°	65°
142°	38°
179°	1°

3. Using algebra to find unknown angles

Often the angles are not given as plain numbers, but as expressions such as $x$ and $2 x$ , or $(x + 10) °$ and $(2 x + 20) °$ . The method is always the same:

1Decide whether the angles are complementary (sum 90°) or supplementary (sum 180°).
2Write an equation: the sum of the two angles equals 90° or 180°.
3Collect like terms and solve for $x$ .
4Substitute back to find each angle, and check that the total is correct.

For instance, suppose two angles $x$ and $2 x$ are complementary. Then $x + 2 x = 90$ , so $3 x = 90$ and $x = 30$ . The angles are 30° and 60°. Check: 30° + 60° = 90° ✓.

Sometimes the question uses wording: 'An angle is three times its supplement.' Let the angle be $θ$ . Its supplement is $180 - θ$ . So $θ = 3 (180 - θ)$ , which gives $4 θ = 540$ and $θ = 135°$ . Check: the supplement is 45°, and 3 × 45° = 135° ✓.

4. Telling which pair is which

Given two angles, add them. If the total is 90° the pair is complementary; if it is 180° the pair is supplementary; if it is anything else the pair is neither. Angles do not have to be touching to be complementary or supplementary. Two angles in different corners of a drawing, such as 40° and 50°, are still complementary because their sum is 90°.

Here is why this matters in real work. A tailor cutting a bias strip, a mason setting a block against a wall, and a surveyor marking a plot all rely on the same facts. If you know one angle of a pair, you can find the other by simple subtraction, with no protractor at all. In an examination, this saves time and gives you an exact answer rather than one read from a drawing.

A useful habit is to write the sum you are using before you calculate: 'sum = 90°' or 'sum = 180°'. Then substitute the numbers. Writing the reason in words, for example 'angles on a straight line add up to 180°', also earns method marks in the BECE.

Test a pair by adding
Pair	Sum	Name
25° and 65°	90°	Complementary
100° and 80°	180°	Supplementary
45° and 45°	90°	Complementary
70° and 30°	100°	Neither
90° and 90°	180°	Supplementary

Worked examples

Try each one yourself first, then open the solution and compare every step.

Example 1

Complement and supplement

Find (a) the complement of 38°, (b) the supplement of 115°.

Show the solution

Step 1(a) Complement = 90° − 38° = 52°.
Step 2(b) Supplement = 180° − 115° = 65°.
Step 3Check: 38° + 52° = 90° ✓ and 115° + 65° = 180° ✓.

Answer: (a) 52° (b) 65°

Example 2

Side road at a junction

A side road joins a straight road at Nkawkaw and makes an angle of 125° on one side. What angle does it make on the other side?

▶ Two angles on a straight line add up to 180°.

Show the solution

Step 1The two angles are on a straight line, so they are supplementary.
Step 2Other angle = 180° − 125° = 55°.
Step 3Check: 125° + 55° = 180° ✓.

Answer: 55°

Example 3

Two angles in terms of x

The angles

(x + 10) °

and

(2 x + 20) °

are supplementary. Find

x

and both angles.

Show the solution

Step 1Supplementary, so the sum is 180°: $(x + 10) + (2 x + 20) = 180$ .
Step 2Collect like terms: $3 x + 30 = 180$ .
Step 3Subtract 30: $3 x = 150$ , so $x = 50$ .
Step 4The angles are $50 + 10 = 60$ and $2 (50) + 20 = 120$ .
Step 5Check: 60° + 120° = 180° ✓.

Answer: $x = 50$ ; the angles are 60° and 120°

Example 4

Complementary with a difference

Two angles are complementary. One is 24° bigger than the other. Find both angles.

Show the solution

Step 1Let the smaller angle be $x$ , so the larger is $x + 24$ .
Step 2Complementary: $x + (x + 24) = 90$ .
Step 3 $2 x + 24 = 90$ , so $2 x = 66$ and $x = 33$ .
Step 4The larger angle is $33 + 24 = 57$ .
Step 5Check: 33° + 57° = 90° ✓ and 57° − 33° = 24° ✓.

Answer: 33° and 57°

Example 5

Complement and supplement together

The complement of an angle is one quarter of its supplement. Find the angle.

Show the solution

Step 1Let the angle be $θ$ . Its complement is $90 - θ$ and its supplement is $180 - θ$ .
Step 2Write the equation: $90 - θ = \frac{180 - θ}{4}$ .
Step 3Multiply both sides by 4: $360 - 4 θ = 180 - θ$ .
Step 4Rearrange: $180 = 3 θ$ , so $θ = 60$ .
Step 5Check: complement = 30°, supplement = 120°, and 120° ÷ 4 = 30° ✓.

Answer: 60°

Summary

Complementary angles add up to 90°; the complement of x is 90° − x.
Supplementary angles add up to 180°; the supplement of x is 180° − x.
Two angles on a straight line are supplementary.
For unknown angles, write an equation using 90° or 180°, solve it, then check the total.
Two acute angles can be complementary; an acute and an obtuse angle can be supplementary; two obtuse angles can never be supplementary.

The carpenter's frame at Suame Magazine

1. Complementary angles: pairs that make a right angle

2. Supplementary angles: pairs that make a straight line

3. Using algebra to find unknown angles

4. Telling which pair is which

Worked examples

Complement and supplement

Side road at a junction

Two angles in terms of x

Complementary with a difference

Complement and supplement together

Summary

Practice: 15 questions