The facts everything else is built from
Angle questions look varied and are all assembled from a handful of facts. Learning these and naming them when you use them is what earns the marks — a mark scheme awards the reason as often as the number.
- Angles on a straight line add to 180°.
- Angles at a point add to 360°.
- Vertically opposite angles are equal.
- Angles in a triangle add to 180°; in a quadrilateral, 360°.
- An exterior angle of a triangle equals the sum of the two opposite interior angles.
- In an isosceles triangle the base angles are equal.
| With parallel lines | Position | Relationship |
|---|---|---|
| Corresponding (F shape) | same side, same position | equal |
| Alternate (Z shape) | opposite sides of the transversal | equal |
| Co-interior (C shape) | same side, between the lines | add to 180° |
Co-interior angles add, they do not match
Corresponding and alternate angles are equal; co-interior angles are supplementary. The C shape is the odd one out, and treating it like the other two is the commonest error with parallel lines. If the two angles are on the same side of the transversal and between the parallels, they add to 180°.
Polygons
A polygon is a closed shape with straight sides. It is regular when all its sides and all its angles are equal — both conditions, since a rhombus has equal sides and unequal angles.
The interior angle sum comes from splitting the polygon into triangles. Joining one vertex to all the others produces n − 2 triangles, each contributing 180°, which is where the formula comes from rather than being a rule to memorise.
Choose Triangles and drag n — the polygon splits into n − 2 triangles, which is the proof of the formula. Now choose Exterior: the exterior sum stays at 360° however many sides there are.
Why the exterior angles always total 360°
Walk once around the outside of the polygon. At each corner you turn through the exterior angle, and by the time you are back where you started facing the same way you have turned through exactly one full revolution. The number of corners makes no difference to that, which is why the sum is 360° for a triangle and for a hundred-sided polygon alike.
Working through an angle problem
The method is to find something you can determine, write it on the diagram with its reason, and repeat. Almost no angle problem is solved in one step, and each intermediate angle is usually worth a mark.
A regular polygon has an interior angle of 156°. How many sides does it have?
- Interior and exterior angles at a vertex lie on a straight line, so the exterior angle is
180 − 156 = 24°.Going via the exterior angle is far quicker than using the interior sum formula. - The exterior angles total 360°, and in a regular polygon they are all equal.This is the fact that does not depend on n.
n = 360 ÷ 24 = 15.- Check with the interior formula:
(15 − 2) × 180 ÷ 15 = 2340 ÷ 15 = 156✓The check uses the other route, so agreement confirms both.
15 sides
Go via the exterior angle
Given an interior angle of a regular polygon, subtract from 180° and divide 360° by the result. Two short steps. Setting up (n − 2) × 180 ÷ n = 156 and solving for n gives the same answer after considerably more algebra, and there is more to go wrong.
Naming and describing shapes
The syllabus expects the properties of the standard quadrilaterals by name, and questions often ask which shape a description fits — so the distinguishing property of each is what to remember.
| Shape | Distinguishing properties |
|---|---|
| Square | four equal sides, four right angles |
| Rectangle | opposite sides equal, four right angles |
| Rhombus | four equal sides, opposite angles equal, diagonals meet at 90° |
| Parallelogram | both pairs of opposite sides parallel and equal |
| Trapezium | exactly one pair of parallel sides |
| Kite | two pairs of adjacent equal sides, one diagonal bisects the other at 90° |
Before you leave this chapter
- Straight line 180°, point 360°, triangle 180°, quadrilateral 360°.
- Corresponding and alternate angles are equal; co-interior angles add to 180°.
- Interior sum is (n − 2) × 180°, from splitting into n − 2 triangles.
- Exterior angles always total 360°, whatever the number of sides.
- Name the reason for every step — the reason is marked as often as the number.
Setting out an angle proof
Geometry questions ask you to "give reasons" and the reasons carry marks in their own right — frequently as many as the numbers. A calculation with no justification typically earns half of what is available.
The convention is to state the value, then the reason, in a fixed form: ∠ABC = 65° (alternate angles). Each line should follow from something already established, so an examiner can read the chain from the given information to the answer without guessing.
- Write each angle on the diagram as you find it — later steps depend on earlier ones.
- Name the reason every time, using the standard wording: "angles on a straight line", "alternate angles", "angles in a triangle", "base angles of an isosceles triangle".
- Where several routes exist, pick the shortest; every extra step is another chance to make an error.
- Check at the end that the angles you have found are consistent — the three in each triangle should still total 180°.
Do not measure the diagram
Exam diagrams are marked "not to scale" precisely so that measuring gives the wrong answer. An angle that looks like a right angle is not one unless the question says so, and two sides that look equal are not equal unless marked. Every fact used must come from the question or be deduced — never from the appearance of the drawing.