Grade by Grade
E. Spatial Sense · Spatial Sense: Shapes, Grids, Angles, Area and Surface Area

Complementary, Supplementary and Opposite Angles

E2.3About 40 minutes
Learning goals
  • I can find the complement or the supplement of an angle.
  • I can explain why opposite angles are equal when two lines cross.
  • I can use angle properties to find unknown angles without measuring.

Words to know

complementary angles
Two angles that add to 90°. Each one is the complement of the other.
supplementary angles
Two angles that add to 180°. Each one is the supplement of the other.
opposite angles
When two straight lines cross, the angles directly across from each other. They are always equal. (They are also called vertically opposite angles.)
adjacent angles
Angles that share a vertex and an arm, side by side.

Let's start with a snowy crossroads

Two straight roads cross outside town. The snowplough driver notices that the corners across from each other always look the same size. Is that a coincidence? Today you'll find out, and you'll learn to find angles without a protractor, just by reasoning.

In the last lesson, you saw that angles that make a full turn add to 360°. Two more totals are worth knowing by heart: a right angle is 90° and a straight angle is 180°.

An aerial view of two snowy country roads crossing in an X, with a snowplough on one road.
Where two straight roads cross, the angles across from each other are opposite angles, and they are always equal. AI-generated image
35°55°complementary35° + 55° = 90°130°50°supplementary130° + 50° = 180°
Left: a right angle split into 35° and 55°, which add to 90°, so they are complementary. Right: a straight line split into 130° and 50°, which add to 180°, so they are supplementary. The angles are drawn to their true size.

Complementary and supplementary

  • Two angles that add to 90° are complementary. To find the complement of an angle, subtract it from 90°.
  • Two angles that add to 180° are supplementary. To find the supplement of an angle, subtract it from 180°.

Memory trick: C comes before S in the alphabet, and 90 comes before 180. Corner (90°), Straight (180°).

Worked example

Example 1: Find the complement

What is the complement of a 28° angle?

  1. Complementary angles add to 90°.
  2. 90° − 28° = 62°.
  3. Check: 28° + 62° = 90° ✓

62°

Worked example

Example 2: Find the supplement

A ladder leans on a wall. It makes a 65° angle with the ground on one side. What angle does it make with the ground on the other side?

  1. The ground is a straight line, and the ladder splits the straight angle into two parts.
  2. The two parts are supplementary: they add to 180°.
  3. 180° − 65° = 115°.

115°

abcda = 40°b = 140°c = 40°d = 140°Opposite angles:a = c and b = d
Two straight lines cross and make four angles: a = 40°, b = 140°, c = 40° and d = 140°. Opposite angles are equal: a = c and b = d.

Why opposite angles are equal

Look at the picture of the crossing lines.

  • Angles a and b sit together on a straight line, so a + b = 180°.
  • Angles b and c also sit together on a straight line, so b + c = 180°.
  • Both a and c are "180° minus b". So a = c!

The same reasoning shows that b = d. This is a proof: it works for any two crossing lines, not just this picture.

Worked example

Example 3: Crossing lines

Two straight lines cross. One of the angles is 72°. Find the other three.

  1. The angle opposite the 72° angle is also 72°.
  2. The angles next to it are supplementary to it: 180° − 72° = 108°.
  3. The angle opposite that one is also 108°.
  4. Check: 72° + 108° + 72° + 108° = 360° ✓ (a full turn).

72°, 108° and 108° (plus the given 72°).

Just for fun

Why did the obtuse angle go to the beach?

Because it was over 90 degrees!

Reason, then check

In these problems you reason instead of measuring. Afterward, you can check with a protractor if there's a drawing. And always check your totals: angles in a corner make 90°, on a straight line 180°, and all the way around a point 360°.

Hands-on activity

Crossing strips

You need: 2 long strips of paper or 2 rulers, a sheet of paper, a pencil and a protractor.

  1. Lay the two strips across each other on the paper to make an X. Trace along both edges of each strip with a pencil, then remove the strips. Use the middle of each traced strip as your line, or simply draw two crossing lines with a ruler.
  2. Label the four angles a, b, c and d, going around.
  3. Measure one angle with the protractor. Use reasoning to predict the other three.
  4. Now measure them to check. Were your predictions within a degree or two?
  5. Draw a new X with a different crossing angle and do it again.
  6. Find a pair of crossing lines at home (window bars, a tic-tac-toe board, the edges of a folding chair) and estimate the angles.

Math talk: "How did you know the opposite angle without measuring?"

Your turn

1

What is the complement of 40°?

2

What is the supplement of 40°?

3

Are 75° and 105° supplementary? Explain.

4

Two straight lines cross. One angle is 125°. Find the other three.

5

Three angles sit side by side on a straight line: 50°, 60° and an unknown angle. Find the unknown angle.

6

A right angle is split into two angles. One is 18°. What is the other?

More practice: warm-up

1

Complementary angles add to how many degrees?

2

Supplementary angles add to how many degrees?

3

What is the complement of 80°?

4

What is the supplement of 90°?

5

When two lines cross, what is true about opposite angles?

More practice: on your own

1

What is the complement of 37°?

2

What is the supplement of 128°?

3

Can an obtuse angle have a complement? Explain.

4

Two straight lines cross, and one angle is 90°. What are the other three angles? What do we call these lines?

5

Angle A and angle B are supplementary. Angle B and angle C are also supplementary. Angle A is 65°. What is angle C? Why?

Challenge

1

Two straight lines cross. One angle is 4 times as big as the angle next to it. Find both angles.

2

An angle is 3 times its complement. Find the angle.

3

Three straight lines cross at one point, making 6 angles. Going around, the first two angles are 50° and 60°. Find all six.

Check your understanding

Quiz

1.

What is the complement of 35°?

2.

What is the supplement of 35°?

3.

Which pair of angles is complementary?

4.

Two straight lines cross. One angle is 110°. What is the angle opposite it?

5.

Two straight lines cross. One angle is 110°. What is the angle next to it?

6.

Two angles on a straight line are x and 47°. What is x?

For parents and teachers

Big idea: The Ontario curriculum (E2.3) asks Grade 6 students to use the properties of supplementary angles, complementary angles, opposite angles, and interior and exterior angles to solve for unknown angle measures. This lesson covers the first three; the next lesson covers interior and exterior angles.

Common difficulties:

  • Mixing up complementary (90°) and supplementary (180°). Use the memory trick: Corner and Straight, in alphabetical order like 90 and 180.
  • Measuring instead of reasoning. Encourage your child to work out angles from the facts first, then check with a protractor.
  • Believing pictures over facts. A sketch may not be drawn to scale; the numbers given are what count.

How to help: Look for crossing lines around the house (window frames, scissors opened partway, road intersections on a map) and ask: "If this angle is about 60°, what are the others?"