Angles at a Point and on a Line

Use angles on a straight line (180°), angles at a point (360°) and vertically opposite angles, with the exact reason, to find missing angles.

In the exam for Cambridge IGCSE Maths 0580, Edexcel IGCSE Maths 4MA1, Cambridge O Level Maths 4024 · 26 real past-paper questions below

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Angles on a Straight Line

Angles that sit side by side on a straight line make half a turn. Together they are $18 0^{\circ}$ .

The 3-mark question: a straight line has angles $10 6^{\circ}$ , $x$ and $x$ . Find $x$ .

Worked example

1
$106 + x + x = 180$ — Angles on a straight line add up to 180°, so the three angles together make 180.
2
$106 + 2 x = 180$ — Collect the two equal angles: x + x = 2x.
3
$2 x = 74$ — Take 106 from both sides, so the x terms are alone: 180 − 106 = 74.
4
$x = 3 7^{\circ}$ — Divide both sides by 2. The two equal angles share the 74 between them.
5
$106 + 37 + 37 = 180$ — Check: the three angles add to 180, so the answer fits the straight line.

Common mistake

Stopping at 74. Two angles share it, so each is half.

Angles at a Point

Angles all the way round a point make a full turn. Together they are $36 0^{\circ}$ .

The 2-mark question: three angles meet at a point. Two are $12 5^{\circ}$ and $9 8^{\circ}$ . Find the third, $x$ .

Worked example

1
$125 + 98 + x = 360$ — Angles at a point add up to 360°, so all three together make a full turn.
2
$223 + x = 360$ — Add the two angles you know first: 125 + 98 = 223.
3
$x = 360 - 223$ — Take 223 from both sides, so x is alone.
4
$x = 13 7^{\circ}$ — The missing angle. Check: 125 + 98 + 137 = 360.

A small square at the point. Exams often mark one angle with a little square instead of a number. That square means $9 0^{\circ}$ , and it must go in your sum.

Worked example

1
$90 + 112 + 75 + w = 360$ — Four angles meet at the point. The square is 90°, so it counts like any other angle.
2
$90 + 112 + 75 = 277$ — Add the three angles you know: 90 + 112 + 75 = 277.
3
$w = 360 - 277 = 8 3^{\circ}$ — Angles at a point add up to 360°, so take 277 from 360.

Key idea

One angle of $14 0^{\circ}$ at a point leaves a reflex angle of $360 - 140 = 22 0^{\circ}$ on the other side.

Common mistake

Leaving out the square. It has no number on it, but it is still 90°.

Vertically Opposite Angles

When two straight lines cross, they make an X. The angles opposite each other in the X are equal.

The 2-mark question: one angle at the crossing is $7 2^{\circ}$ . Find the other three.

Worked example

1
$opposite = 7 2^{\circ}$ — The angle across the crossing point faces the 72°. Vertically opposite angles are equal.
2
$next to it = 180 - 72 = 10 8^{\circ}$ — The 72° and its neighbour sit on one straight line. Angles on a straight line add up to 180°.
3
$last one = 10 8^{\circ}$ — It faces the 108° across the crossing. Vertically opposite angles are equal.
4
$72 + 108 + 72 + 108 = 360$ — Check: the four angles go all the way round the point, so they must make 360°.

Common mistake

Thinking "vertically" means up and down. It means opposite at the vertex, the point where the lines cross.

Angles Written With x

Exams often write each angle with x in it, like $(3 x + 15)^{\circ}$ . Use the same angle fact to make an equation, then solve it.

On a straight line. Two angles on a straight line are $(3 x + 15)^{\circ}$ and $(2 x - 5)^{\circ}$ . Find both angles.

Worked example

1
$(3 x + 15) + (2 x - 5) = 180$ — Angles on a straight line add up to 180°, so add the two expressions and make them equal 180.
2
$5 x + 10 = 180$ — Collect like terms: 3x + 2x = 5x and 15 + (−5) = 10. Keep the minus with the 5.
3
$5 x = 170$ — Take 10 from both sides, so the x term is alone.
4
$x = 34$ — Divide both sides by 5. This is x, not an angle yet.
5
$3 (34) + 15 = 11 7^{\circ}$ — Put x = 34 back into the first expression to get the first angle.
6
$2 (34) - 5 = 6 3^{\circ}$ — Put x = 34 into the second expression to get the other angle.
7
$117 + 63 = 180$ — Check: they add to 180, and both are positive, so they can be real angles.

Vertically opposite. Two vertically opposite angles are $(4 y - 20)^{\circ}$ and $(2 y + 30)^{\circ}$ . Find y.

Worked example

1
$4 y - 20 = 2 y + 30$ — Vertically opposite angles are equal, so the two expressions are equal. Do not add them to 180.
2
$2 y - 20 = 30$ — Take 2y from both sides, so y is on one side only.
3
$2 y = 50$ — Add 20 to both sides.
4
$y = 25$ — Divide both sides by 2.
5
$4 (25) - 20 = 80 = 2 (25) + 30$ — Check: both expressions give 80°, so they really are equal.

Common mistake

Giving x = 34 as the answer when the question asks for the angles. Put x back in.

Key idea

Show your equation and each step. The equation alone earns a mark, even if a slip later gives the wrong angle. If an angle comes out 0 or negative, the working has a mistake: an angle in a diagram is always more than 0°.

Angles in a Ratio

Sometimes you are told the angles are in a ratio, like $2 : 3 : 4$ . Share the full total, 180° or 360°, in that ratio.

Three angles $a$ , $b$ and $c$ meet at a point. $a : b : c = 2 : 3 : 4$ . Find $c$ .

Worked example

1
$2 + 3 + 4 = 9$ — Add the ratio numbers. The full turn is cut into 9 equal parts.
2
$360 \div 9 = 40$ — Angles at a point add up to 360°, so one part is 40°.
3
$a = 2 \times 40 = 8 0^{\circ}$ — a has 2 parts.
4
$b = 3 \times 40 = 12 0^{\circ}$ — b has 3 parts.
5
$c = 4 \times 40 = 16 0^{\circ}$ — c has 4 parts. This is the one the question asks for.
6
$80 + 120 + 160 = 360$ — Check: the three angles make a full turn.

Common mistake

Sharing 180 when the angles go all the way round a point. Use 360 for a point and 180 for a straight line.

Angles on a Clock Face

A clock face is a full turn, so it is a question about angles at a point. The 12 numbers split the 360° into equal gaps.

Rule

Each gap between two numbers is $360 \div 12 = 3 0^{\circ}$ .
The hour hand moves 30° in one hour, so it moves half a gap in half an hour.

Find the reflex angle between the hands at 4.30.

Worked example

1
$360 \div 12 = 3 0^{\circ}$ — 12 equal gaps share the full turn, so each gap is 30°.
2
$minute hand on 6$ — At half past, the minute hand points straight down at the 6.
3
$hour hand at 4 \frac{1}{2}$ — Half an hour has passed since 4 o'clock, so the hour hand is halfway between 4 and 5, not on the 4.
4
$1 \frac{1}{2} \times 30 = 4 5^{\circ}$ — From $4 \frac{1}{2}$ to 6 is one and a half gaps. This is the small angle between the hands.
5
$360 - 45 = 31 5^{\circ}$ — The reflex angle goes the long way round. Angles at a point add up to 360°.

Common mistake

Putting the hour hand exactly on the 4. That gives 60°, but by 4.30 the hour hand has moved halfway to the 5.

Which Method? Read the Question

Look at what the angles are sitting on.

You see	You do
Angles along one straight line	180 − the rest
Angles all round a point	360 − the rest
Two lines crossing like an X	opposite angles are equal
Two equal angles x	subtract, then halve
Angles written with x	make an equation, solve, put x back in
Angles in a ratio	share 180 or 360 in the ratio
Parallel lines too	Angles in Parallel Lines
A triangle too	Angles in a Triangle

Rule

"Give a reason" earns its own mark. Write one of these sentences in full:

Angles on a straight line add up to 180°.
Angles at a point add up to 360°.
Vertically opposite angles are equal.

Key idea

One word like "line" or "opposite" may not score. Say what the angles do: add up to 180°, add up to 360°, or are equal.

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