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.
Also known as: vertical angles · vertically opposite angles · adjacent angles on a straight line
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°
- 1Write the statement: 130° + x = 180°.Reason: angles on a straight line.
- 2x = 180° − 130°.Subtract to isolate x.
- 3x = 50°.Always include the degree symbol.
2Angles round a point add to 360°
- 1120° + 150° + y = 360°.Reason: angles round a point.
- 2270° + y = 360°.Add the known angles first.
- 3y = 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.
- 1The vertically opposite angle is also 65°.Reason: vertically opposite angles are equal.
- 2A neighbouring angle: 180° − 65° = 115°.Reason: angles on a straight line.
- 3Its 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.
- 135° + z = 90°.Reason: the lines are perpendicular, so the full angle is 90°.
- 2z = 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.
- 1Two angles on a straight line are (3x + 5)° and (x + 15)°. Determine the value of x, and hence the size of each angle.
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Angles on a straight line add up to 180° (1)
4x + 20 = 180 (1)
x = 40 (1)
The angles are 125° and 55° (1) - 2Two vertically opposite angles are (4x + 10)° and (2x + 50)°. Determine the value of x, and hence the size of each angle.
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Vertically opposite angles are equal (1)
2x = 40 (1)
x = 20 (1)
The angles are 90° and 90° (1) - 3Angles of 70° and x lie on a straight line. Find x.
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180° − 70° = 110° - 4Angles of 95° and y lie on a straight line. Find y.
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180° − 95° = 85° - 5Angles of 140°, 90° and p meet at a point. Find p.
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360° − 230° = 130° - 6Two lines cross forming an angle of 48°. What is its vertically opposite angle?
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48° (vertically opposite angles are equal) - 7Two lines cross forming an angle of 48°. What is the neighbouring angle?
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180° − 48° = 132° - 8A right angle is split into 28° and k. Find k.
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90° − 28° = 62° - 9What does AB ⊥ CD mean?
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AB is perpendicular to CD — they meet at 90°. - 10Angles 3x and 2x lie on a straight line. Find x.
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3x + 2x = 180 → 5x = 180 → x = 36°
Quick Quiz
4 quick questions on what you just read. Take it when you feel ready.
Now practise it
Download Grade 8 past papers and worksheets on this topic.