Angle Facts
Every time a door swings open, a clock hand moves or a skateboarder spins, something turns. An angle measures how much. A handful of simple facts about angles (they all come from "a full turn is 360°") let you work out angles you can't measure, and explain exactly why you're right.
National curriculum KS3: draw and measure angles; apply the properties of angles at a point, angles at a point on a straight line and vertically opposite angles; understand and use alternate and corresponding angles on parallel lines; derive and use the sum of angles in a triangle and use it to deduce the angle sum in any polygon, and to derive properties of regular polygons1 · What an angle is: an amount of turn
Hold a pencil flat on the table, keep one end still and spin the other end round. The amount the pencil has turned is an angle. Nothing else matters: not how long the pencil is, only how far it has turned.
When we draw an angle we show two straight lines that meet at a point. The point where they meet is called the vertex (plural: vertices). The two lines are called the arms. One arm is where the turn starts, the other is where it stops. A little curved arc between the arms shows which gap we mean.
We measure angles in degrees, written with a small circle: °. A whole turn, all the way round back to the start, is split into 360 equal steps, so one full turn is 360°. (People think the number 360 comes from ancient Babylon, and it's handy because 360 divides exactly by 2, 3, 4, 5, 6, 8, 9, 10, 12 and lots more.) From that one fact:
- half a turn is 360 ÷ 2 = 180° (you end up facing backwards, and the two arms make a straight line);
- a quarter turn is 360 ÷ 4 = 90° (the corner of a page, a "square corner");
- three-quarters of a turn is 3 × 90 = 270°.
Drag the round handle (or use the slider) to turn the arm. The scale round the outside works like a protractor, the see-through tool you use to measure angles.
Using a real protractor. Put the centre cross exactly on the vertex. Line up the 0° line with one arm. Then count round from that 0 to the other arm. A protractor has two scales of numbers, one going each way. The most common mistake is reading the wrong one: if the angle is clearly smaller than a square corner but you read 130°, you've used the wrong scale (the right answer would be 180 − 130 = 50°). Always ask: "is it smaller or bigger than 90°?" before you read.
2 · Types of angle
Angles get names depending on how big they are. You only need two landmarks to sort any angle: the square corner (90°) and the straight line (180°).
| Name | Size | How to spot it |
|---|---|---|
| Acute | between 0° and 90° | Sharp and pointy. It fits inside the corner of a page. ("A-cute little angle.") |
| Right angle | exactly 90° | A square corner. Always marked with a small square instead of an arc. Two lines that meet at a right angle are called perpendicular. |
| Obtuse | between 90° and 180° | Blunt and wide open, but not yet flat. |
| Straight angle | exactly 180° | The two arms make one straight line. |
| Reflex | between 180° and 360° | More than a straight line: the arc goes the "long way round" the outside. |
Every pair of arms actually makes two angles: the small one inside and the big one round the outside. Together they make a full turn, so if the inside is 70°, the reflex angle outside is 360 − 70 = 290°. The arc tells you which one is meant.
Name that angle: what type is the orange one?
Estimating tip: first decide which landmarks the angle sits between (0°, 90°, 180°, 270°, 360°), then ask "about halfway? closer to one end?". Half a right angle is 45°, which is a useful extra landmark.
3 · Angles on a straight line and angles at a point
Here's the first angle fact. If several angles sit next to each other along a straight line, all sharing the same vertex, then together they make half a turn. So:
angles on a straight line add up to 180°
And if angles go all the way round a point with no gaps, together they make a full turn:
angles at a point add up to 360°
Neither of these is a new rule to memorise: they're just "half a turn = 180°" and "a full turn = 360°". Drag the handles and watch the total stay the same no matter what you do.
Try it: find x
Common mistake: using the wrong total. Look at the diagram before you calculate. Is there a straight line with all the angles on one side of it? Use 180°. Do the angles go all the way round? Use 360°. Mixing these up is the single most common error in angle questions.
Another trap: diagrams in tests are usually "not drawn accurately", so you must calculate, not measure with a protractor.
4 · Vertically opposite angles
When two straight lines cross, they make an X shape with four angles at the crossing point. The angles that are across from each other (not next to each other) are called vertically opposite angles. ("Vertically" here is about the shared vertex, not about up-and-down.)
The fact: vertically opposite angles are equal. Drag either line and watch the orange angles match each other, and the plum angles match each other.
Why it's true. Look at angles a and b: they sit together on one straight line, so a + b = 180°. Now look at b and c: they sit together on the other straight line, so b + c = 180°. Both a and c are "whatever is left when you take b away from 180°", so they must be the same. The working box above does this with the actual numbers.
Watch out: the angles must be made by two straight lines crossing. If a line bends at the vertex, the opposite angles are not equal. And the neighbouring angles (like a and b) are not equal: they add up to 180° instead.
5 · Angles in a triangle and a quadrilateral
A triangle is a flat shape with three straight sides and three corners. Each corner has an angle inside the shape, called an interior angle. Here's a surprising fact: whatever triangle you draw, long and thin or fat and pointy, the three interior angles always add up to exactly the same total.
Try the classic experiment. Cut out a paper triangle, tear off its three corners and lay them side by side with their points together. They always fit perfectly along a straight line. Do it here: drag the corners to make any triangle you like, then press Tear off next corner.
Try it: find the missing angle
angles in a triangle add up to 180°
A quadrilateral is any flat shape with four straight sides ("quad" means four, "lateral" means sides): squares, rectangles, kites and wonky ones too. Switch the widget to quadrilateral. The dashed line, called a diagonal, joins two opposite corners and cuts the quadrilateral into two triangles. Each triangle's angles add up to 180°, so altogether 2 × 180° = 360°. Tear the four corners off and they fit round a point: a full turn.
angles in a quadrilateral add up to 360°
Two special triangles come up all the time. In an equilateral triangle all three sides are equal and so are all three angles: 180 ÷ 3 = 60° each. In an isosceles triangle two sides are equal (shown by little dashes on the sides), and the two angles at the bottom of those sides, the base angles, are equal too.
Common mistakes: using 360° for a triangle (that's the quadrilateral total), and forgetting that a right-angle square symbol means one of the angles is 90° even though no number is written. In a triangle, if you already know two angles, the third is 180 minus their total, never 180 minus just one of them.
6 · Angles on parallel lines
Two lines are parallel if they go in exactly the same direction, so they stay the same distance apart and never meet, like railway tracks or the lines on lined paper. In diagrams parallel lines are marked with matching little arrows (>). A line that cuts across both of them is called a transversal.
The transversal makes four angles at each crossing: eight angles in total. But because the parallel lines point the same way, the transversal crosses each one in exactly the same way. That gives three fact pairs. The glowing lines trace the shape that helps you spot each one:
| Pair | Where they are | Shape | Rule |
|---|---|---|---|
| Corresponding | In the same position at each crossing (both top-right, say) | F | equal |
| Alternate | Between the parallel lines, on opposite sides of the transversal | Z | equal |
| Co-interior | Between the parallel lines, on the same side of the transversal | C or U | add up to 180° |
Choose a pair type, then tilt the transversal by dragging anywhere on the diagram or using the slider. The letters a to h name all eight angles. Other pair shows the other examples of the same type.
Try it: find x and give the reason
Why co-interior angles add to 180°. In the diagram, angle d and angle e are co-interior. Angle e is equal to angle a (corresponding). And a + d = 180° because they sit together on a straight line. So d + e = d + a = 180°.
Common mistakes: using these facts when the lines are not marked parallel (without the arrows, none of these rules apply), and mixing up "equal" with "add to 180°". Quick sense check: on parallel lines every angle is one of just two sizes, a small one and a big one, which add up to 180°. If x looks the same size as the given angle, they're equal; if one looks acute and the other obtuse, they add up to 180°.
7 · Angles in polygons
A polygon is any flat shape made of straight sides: triangle (3 sides), quadrilateral (4), pentagon (5), hexagon (6), heptagon (7), octagon (8), nonagon (9), decagon (10)… A regular polygon has all its sides the same length and all its angles the same size, like a stop sign (a regular octagon) or the cells in a honeycomb (regular hexagons).
Interior angles: split it into triangles
Pick one corner and draw lines from it to every other corner you can. The polygon splits into triangles. A shape with n sides always splits into n − 2 triangles (try it: a pentagon gives 3, a hexagon gives 4). Each triangle gives 180°, so:
sum of interior angles = (n − 2) × 180°
For a regular polygon all the interior angles are equal, so each one is that total divided by n.
Exterior angles: walk a pencil round the edge
Now make one side longer, past the corner (the dashed line). The angle between that extended side and the next side is called an exterior angle. It's exactly the amount you have to turn at that corner if you walk round the edge of the shape. The interior angle and exterior angle at a corner sit together on a straight line, so interior + exterior = 180°.
Put a pencil on the first side and walk it round the shape, turning at every corner. When it gets back to the start it is pointing the same way it began. It has turned exactly once. So, for any polygon that doesn't dent inwards:
exterior angles add up to 360°
For a regular polygon the turns are all equal, so each exterior angle = 360° ÷ n. This is usually the quickest route: a regular decagon turns 360 ÷ 10 = 36° at each corner, so each interior angle is 180 − 36 = 144°. It also works backwards: if each exterior angle is 30°, there must be 360 ÷ 30 = 12 sides.
Try it: polygon practice
Common mistakes: doing n × 180 instead of (n − 2) × 180; mixing up interior and exterior angles (the exterior is the small turn, the interior is the angle inside); and dividing 360 by the interior angle to find the number of sides. Only the exterior angle divides into 360.
8 · Giving reasons
In maths, getting the number isn't the whole job: you have to say which fact you used. A reason is a short sentence naming the angle fact. These are the standard ones. Learn the wording, because "Z-angles" or "because it's parallel" won't get the marks.
| Situation | Reason to write |
|---|---|
| Angles together along a straight line | Angles on a straight line add up to 180° |
| Angles all the way round a point | Angles at a point add up to 360° |
| Two straight lines cross | Vertically opposite angles are equal |
| The three angles of a triangle | Angles in a triangle add up to 180° |
| The four angles of a quadrilateral | Angles in a quadrilateral add up to 360° |
| Two equal sides in a triangle | Base angles of an isosceles triangle are equal |
| Parallel lines, F-shape | Corresponding angles are equal |
| Parallel lines, Z-shape | Alternate angles are equal |
| Parallel lines, C-shape | Co-interior angles add up to 180° |
Harder questions need two or three facts in a row. Write one line per step: the calculation, then the reason. Work through this one. The top and bottom lines are parallel; P is on the top line and Q and R are on the bottom line.
Notice something clever in that diagram: the 50°, c and 60° at P sit together on the top straight line, so they add to 180°. But a = 50° and b = 60° are the triangle's other two angles. So the triangle's three angles add up to 180° too. That's a proof that every triangle's angles add to 180°, using alternate angles, which is even better than tearing paper.
Worked examples
Angles at a point are 90°, 140°, 75° and y. Find y.
An isosceles triangle has a 40° angle between its two equal sides. Find the other two angles.
Parallel lines: one angle is 112°. Find x and y, giving reasons.
Find the size of each interior angle of a regular octagon.
Each interior angle of a regular polygon is 156°. How many sides does it have?
A quadrilateral has angles x, 2x, 100° and 80°. Find x.
Quick check
What you've learned
- An angle measures an amount of turn. A full turn is 360°, half a turn 180°, a quarter turn 90°. The meeting point is the vertex and the two lines are the arms.
- Types: acute (less than 90°), right (exactly 90°, marked with a square), obtuse (between 90° and 180°), straight (180°), reflex (between 180° and 360°).
- Angles on a straight line add up to 180°. Angles at a point add up to 360°.
- When two straight lines cross, vertically opposite angles are equal, and neighbouring angles add to 180°.
- Angles in a triangle add up to 180°; angles in a quadrilateral add up to 360° (two triangles). Isosceles triangles have two equal base angles; equilateral triangles have three 60° angles.
- On parallel lines: corresponding angles (F) are equal, alternate angles (Z) are equal, co-interior angles (C) add up to 180°.
- Polygons: interior angle sum = (n − 2) × 180°; exterior angles add up to 360°; interior + exterior = 180° at each corner. For a regular polygon, each exterior angle = 360° ÷ n and n = 360° ÷ exterior angle.
- Always calculate rather than measure, and write the reason for every step using the standard wording.