Angle Relationships: Straight, Intersecting & Perpendicular Lines

Angles on a straight line, angles round a point, vertically opposite angles and perpendicular lines — the four relationships every geometry rider is built from, with a diagram, worked examples and a quiz.

Geometry in Grade 8 stops being about measuring with a protractor and starts being about reasoning: working out an unknown angle from the ones you already know, and giving a reason for every step. Everything begins with the four relationships below.

1Angles on a straight line add to 180°

Angles on a straight line are supplementary: they add up to 180°.

Worked ExampleFind x if angles of 130° and x° lie on a straight line

1
Write the statement: 130° + x = 180°.
Reason: angles on a straight line.
2
x = 180° − 130°.
Subtract to isolate x.
3
x = 50°.
Always include the degree symbol.

2Angles round a point add to 360°

Angles that meet at a point and fill the whole turn add up to 360°.

Worked ExampleThree angles at a point are 120°, 150° and y. Find y

1
120° + 150° + y = 360°.
Reason: angles round a point.
2
270° + y = 360°.
Add the known angles first.
3
y = 90°.

3Vertically opposite angles are equal

When two straight lines cross, they form four angles. The angles opposite each other at the crossing point are always equal.

Two intersecting lines: a and c are vertically opposite (equal), and so are b and d. Neighbouring angles like a and b lie on a straight line, so they add to 180°.

Worked ExampleTwo lines cross. One angle is 65°. Find the other three

1
The vertically opposite angle is also 65°.
Reason: vertically opposite angles are equal.
2
A neighbouring angle: 180° − 65° = 115°.
Reason: angles on a straight line.
3
Its vertically opposite angle is also 115°.
The four angles are 65°, 115°, 65°, 115° — and they total 360°.

4Perpendicular lines

Two lines are perpendicular when they meet at exactly 90°. This is shown with a small square at the corner, and written AB ⊥ CD.

Worked ExampleAB ⊥ CD. A ray splits the right angle into 35° and z. Find z

1
35° + z = 90°.
Reason: the lines are perpendicular, so the full angle is 90°.
2
z = 55°.

💡

In every geometry answer, write the reason next to your statement — "angles on a straight line", "vertically opposite angles". In CAPS exams the reason carries its own mark.

Practice exercises

Work each one out, then click to reveal the answer.

📄 Worksheet ✅ Memo 📝 Exam paper 🗝️ Marking guide

1
Angles of 70° and x lie on a straight line. Find x.
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180 − 70 = 110°
2
Angles of 95° and y lie on a straight line. Find y.
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180 − 95 = 85°
3
Angles of 140°, 90° and p meet at a point. Find p.
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360 − 230 = 130°
4
Two lines cross forming an angle of 48°. What is its vertically opposite angle?
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48° (vertically opposite angles are equal)
5
Two lines cross forming an angle of 48°. What is the neighbouring angle?
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180 − 48 = 132°
6
A right angle is split into 28° and k. Find k.
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90 − 28 = 62°
7
What does AB ⊥ CD mean?
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AB is perpendicular to CD — they meet at 90°.
8
Angles 3x and 2x lie on a straight line. Find x.
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3x + 2x = 180 → 5x = 180 → x = 36°