GCSE · Maths · AQA · Spec 8300 · Foundation
Angle properties at a point and on a line
Spin all the way round on the spot: that's 360°. Turn to face the opposite way: that's half of it, 180°. Those two turns are the whole of this topic.
Half turns and full turns
The totals never move
∠AOP + ∠POB = 180° and ∠RQS + ∠SQT + ∠TQR = 360°
On the left, drag P round O, above or below the straight line AB. On the right, slide T up and down while QR and QS stay put. The separate angles keep changing. Add them up each time and see what doesn't.
Maths · Algebra
Why opposite angles are twins
Now cross two straight lines, like the letter X, or a pair of open scissors. Going round the crossing, call the four angles a, b, c and d. Angles a and c face each other across the crossing point, so they are called vertically opposite angles. So are b and d. Step through to see why a and c have to be equal.
a and b sit side by side along one of the two straight lines, so together they make a half turn.
Step 1 of 5
a and b sit side by side along one of the two straight lines, so together they make a half turn.
Choose your fact
Which fact finds x?
Read each situation. Which ONE angle fact would you use to find x?
Still to sort
Angles at a point add up to 360° (0)
The angles go all the way round the point.
Where the line is: If the angles only fill ONE side of a straight line, that's a half turn, not a full one.
Angles on a straight line add up to 180° (0)
The angles sit side by side along one side of a straight line.
Where the line is: Two angles next to each other at a crossing belong here too: they share one of the straight lines.
Vertically opposite angles are equal (0)
Two straight lines cross, and the angles face each other across the crossing point.
Where the line is: Only the angle directly across the crossing is equal. The one next door adds up to 180° with it instead.
The hardest part of most angle questions is spotting which fact you need. The arithmetic is the easy bit.
WHAT YOU'VE LEARNED
A quick recap of today's lesson.
Half a turn is 180°, a whole turn is 360°, and every angle fact here comes from those two turns.
What you need to know
- Angles at a point add up to 360°, because together they make one full turn.
- Angles on a straight line add up to 180°, because a straight line is half a turn.
- Vertically opposite angles are equal: when two straight lines cross, each angle makes 180° with the same neighbouring angle as the one facing it.
- When you work out an angle, say which fact you used. The reason is part of the answer.
The big picture
Angles that go all the way round a point make one full turn, so they add up to 360°. Angles along one side of a straight line make half a turn, so they add up to 180°. Where two straight lines cross, the angles facing each other across the crossing are equal, because each one makes 180° with the same neighbour.
Key points
Worked example
Problem
Four angles fill the space all the way round a point: 3x, 2x, 90° and x + 30°. Work out x, then find the size of the largest angle.
⚠ Watch out
Using 180° for angles round a point, or 360° for angles on a straight line. Before you calculate, ask one question: do these angles go all the way round, or only along one side of a straight line?
Memory hook
Straight line = half a spin = 180°. Point = a full spin = 360°. Crossing straight lines work like scissors: however wide you open the blades, the handles open by exactly the same angle.
Check yourself
Cover the page. Using the word 'turn', explain where the totals for angles at a point and on a straight line come from. Then explain why opposite angles at a crossing match.
Flashcards
(11)Angles at a point add up to…?
Angles on a straight line add up to…?
Two straight lines cross. Which angles are equal?
Why is a straight line worth 180°?
Why are vertically opposite angles equal?
What does 'vertically' mean in 'vertically opposite'?
At a crossing, what do two angles right next to each other add up to?
Two straight lines cross at right angles. What are the four angles?
Point total or straight-line total? The one question to ask.
Does adding more angles on a straight line change their total?
You've solved for x in two vertically opposite expressions. How do you check?
Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.
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