Angle relationships

Some angle pairs always add to 90° or 180°, and some are always equal. Spot the pair, and the missing angle follows.

The idea: angles that belong together

When lines meet or cross, the angles they make are linked. If you know one angle, you can often find all the others without a protractor.

There are only a few rules to learn. Each one comes from a right angle (90°), a straight line (180°) or a full turn (360°) (7.G.B.5).

See it: pictures to draw with your child

Copy these onto paper or a whiteboard. A quick sketch makes the idea concrete before any written method.

Complementary pairs make 90°, supplementary pairs make 180°, and vertical angles are equal.

Parallel lines and a transversal: every angle is either 60° or 120°.

The main angle pairs

Pair	Rule	Example
Complementary	Add to 90°	35° and 55°
Supplementary	Add to 180°	60° and 120°
Adjacent on a straight line	Add to 180°	A ray splits a straight line into 70° and 110°
Vertical (vertically opposite)	Equal	Two crossing lines: opposite angles are both 120°
Around a point	Add to 360°	100°, 130° and 130°

A memory trick: C comes before S in the alphabet, and 90 comes before 180. Complementary is 90, supplementary is 180.

Parallel lines cut by a transversal

A transversal is a line that crosses two parallel lines. It makes eight angles, but only two different sizes. Every angle is either the small one or its partner that makes 180°.

Corresponding angles sit in the same position at each crossing. They are equal: look for an F shape.
Alternate interior angles are between the parallel lines, on opposite sides of the transversal. They are equal: look for a Z shape.
Co-interior angles are between the lines, on the same side. They add to 180°: look for a C or U shape.

Angle explorer

Slide to make an angle. What angle would make it up to 90° or to 180°?

Solving for x

Many questions give the angles as expressions. Write the rule as an equation, then solve it.

Two supplementary angles are x and 2x + 30. So x + 2x + 30 = 180, which gives 3x = 150 and x = 50. The angles are 50° and 130°. See equations for more on solving.

Angle relationships in real life

Railway tracks, road markings and the rungs of a ladder are parallel lines. A road that crosses them is a transversal, and the angles repeat at each crossing.

Carpenters use the same rules to cut matching angles for picture frames. Two 45° cuts make a 90° corner.

Key words

Complementary: two angles that add up to 90°.
Supplementary: two angles that add up to 180°.
Vertical angles: opposite angles where two lines cross; they are equal.
Transversal: a line that crosses two or more other lines.
Corresponding angles: angles in matching positions at each crossing.

A week of practice

Ten minutes a day is plenty. Short, regular practice beats one long session at the weekend.

Day 1: complementary and supplementary pairs with the angle explorer.
Day 2: vertical angles: draw two crossing lines and measure all four.
Day 3: parallel lines and a transversal: find F, Z and C shapes.
Day 4: solve for x in angle equations.
Day 5: the quiz on this page, then a round of missing angles practice.

If one day goes badly, repeat it rather than moving on. Each day builds on the one before.

Quiz: find the missing angle

Which rule fits? 90, 180, or equal?

Where this fits: geometry step by step

What to say to your child

A short script for teaching it the first time. Pause after each line and let your child do the next step.

“What does a right angle measure?” (90°.)
“So what goes with 30° to make a right angle?” (60°.)
“What does a straight line measure?” (180°.)
“What goes with 110° on a straight line?” (70°.)
“Two lines cross. What do you notice about the opposite angles?” (They are equal.)
“Can you see an F shape here? What does that tell you?” (The angles are equal.)

More worked examples

Example 1: Complement of 28°

90 − 28 = 62°.

Example 2: Supplement of 135°

180 − 135 = 45°.

Example 3: Vertical angles

One angle is 72°, so the opposite angle is 72°.

Example 4: Co-interior angles

One is 65°, so the other is 180 − 65 = 115°.

Practice questions (answers below)

Ask these out loud or write them down. Check each answer, and ask “how did you get that?” for one or two.

Complement of 40°
Supplement of 40°
Complement of 75°
Supplement of 99°
Vertical angle to 58°
Angles around a point: 120°, 150° and ?
Corresponding angle to 70°
Co-interior angle to 70°
x + 3x = 180. Find x.
Stretch: x and x + 20 are complementary. Find both angles.

Show the answers

50°
140°
15°
81°
58°
90°
70°
110°
45
35° and 55°

Common mistakes

Mixing up complementary and supplementaryUses 180 for complementary.Fix: C before S, 90 before 180.

Thinking all eight transversal angles are equalGives 60° for every angle.Fix: there are two sizes: the angle and 180 minus it.

Using co-interior angles as equalSets them equal instead of adding to 180.Fix: angles between the lines on the same side make 180°.

How to teach it at home

Arm angles

Make a right angle with your arms, then split it with a pencil. Name both parts.

Street spotting

Look for parallel lines and a road crossing them. Point out matching angles.

Scissor angles

Open scissors and look at the four angles. Which are equal?

By grade

Grade 7 (7.G.B.5): supplementary, complementary, vertical and adjacent angles, and solving simple equations for an unknown angle.
Grade 8 (8.G.A.5): angles made when parallel lines are cut by a transversal, and angle facts about triangles.
In India, Parallel and Intersecting Lines (class 7) and Lines and Angles (class 6) cover these rules.
Next: angles in a triangle.

More for each grade: Grade 7 math and the CBSE class 7 maths guide; Grade 8 math and the CBSE class 8 maths guide.