Key Stage 3 maths · Geometry and measures

KS3 angle facts, parallel lines and polygons: a parent’s guide

Angle facts let pupils work out an unknown angle without measuring it. In Years 7 to 9 they use angles at a point, on a straight line and vertically opposite, the rules for parallel lines, the 180° angle sum of a triangle, and from that the angle sum of any polygon and the properties of regular polygons. Just as important, they learn to give a reason for every step.

Angles, parallel lines and polygons, explained

A few facts underpin everything. Angles at a point add up to 360°, because a full turn is 360°. Angles on a straight line add up to 180°. When two straight lines cross, the angles opposite each other, called vertically opposite angles, are equal. These are not new to many pupils, but at Key Stage 3 they are used as reasons: “angles on a straight line add to 180°, so x = 180 − 125 = 55°”. Teachers expect pupils to write such statements, because they train the habit of justifying each step.

When a straight line crosses two parallel lines, the angles it makes are linked. Corresponding angles are in matching positions at the two crossings and are equal (sometimes taught as an F shape). Alternate angles lie on opposite sides of the crossing line, between the parallel lines, and are also equal (a Z shape). Co-interior angles lie between the parallel lines on the same side and add to 180° (a C or U shape). These letter shapes are memory aids, but the key is the reason: pupils must be able to say “alternate angles” and know that the lines have to be parallel, shown by arrows on the diagram.

The angles in any triangle add up to 180°. Pupils see this by tearing off the three corners and placing them side by side to form a straight line, and later prove it using parallel lines. From there they deduce the angle sum of other polygons. Choose one corner of a polygon and draw lines to every other corner it is not next to: this splits the shape into triangles. A quadrilateral makes 2 triangles (360°), a pentagon 3 triangles (540°), and in general a polygon with n sides makes n − 2 triangles, so the angle sum is (n − 2) × 180°.

Regular polygons have equal sides and equal angles. Each interior angle is the angle sum divided by the number of sides. A different route uses exterior angles, which are the angles you turn through at each corner when walking around the outside. Together they make one full turn, 360°, so each exterior angle of a regular polygon is 360° ÷ n, and the interior angle is 180° minus that. Pupils use whichever method suits the question and check one against the other.

Words your child will hear

Vertically opposite angles: The pairs of equal angles opposite each other where two straight lines cross.
Alternate angles: Equal angles on opposite sides of a line crossing two parallel lines, forming a Z shape.
Corresponding angles: Equal angles in matching positions at each crossing of a line with two parallel lines, forming an F shape.
Co-interior angles: Angles between parallel lines on the same side of the crossing line. They add up to 180°.
Interior angle: An angle inside a polygon, at one of its corners.
Exterior angle: The angle between one side of a polygon and the extension of the next side. Its partner interior angle makes 180° with it.
Regular polygon: A polygon with all sides equal and all angles equal, such as a square or a regular hexagon.

What the National Curriculum says

In Key Stage 3, pupils should be taught to:

Apply the properties of angles at a point, angles at a point on a straight line, vertically opposite angles
Understand and use the relationship between parallel lines and alternate and corresponding angles
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 polygons

Objectives from the National Curriculum in England, © Crown copyright, used under the Open Government Licence v3.0.

A worked example

Find the size of each interior angle of a regular decagon (a polygon with 10 sides). Show it two ways.

Method 1: angle sum. Choose one corner and draw lines to the corners it is not next to. A decagon splits into 10 − 2 = 8 triangles, each with an angle sum of 180°.
```
Sum = (10 − 2) × 180
    = 8 × 180
    = 1,440°
```
In a regular polygon all 10 interior angles are equal, so divide the total by 10.
```
1,440 ÷ 10 = 144°
```
Method 2: exterior angles. The exterior angles of any polygon add up to 360°. In a regular decagon there are 10 equal ones.
```
Exterior = 360 ÷ 10 = 36°
```
An interior angle and its exterior angle lie on a straight line, so they add up to 180°.
```
Interior = 180 − 36 = 144°
```
Both methods agree, which is a good check on the arithmetic.

Each interior angle of a regular decagon is 144°.

Common mistakes and how to help

Assuming lines are parallel because they look parallel

Diagrams in maths questions are often marked “not drawn accurately”, so lines that look parallel by eye may not be. Only arrow marks, or the wording of the question, tell you they are.

What helps: Only use alternate or corresponding angle rules when the question says or marks that the lines are parallel. Ask your child, “How do you know?”

Giving an answer with no reason

Pupils who can see the answer jump to it, but a bare number earns little credit when marks are given for reasoning.

What helps: Practise the sentence pattern: “Angles on a straight line add to 180°, so …”. Reading the reason aloud before writing the sum is a useful habit.

Using 360 ÷ n as the interior angle

The number 360 ÷ n is the exterior angle, but the pupil remembers “divide 360 by the number of sides” and stops.

What helps: Draw a regular pentagon and mark one interior angle and its exterior angle. The exterior angle is the turn at the corner. Subtract it from 180° to get the interior angle.

Using n × 180 or (n − 1) × 180 for the angle sum

The rule (n − 2) × 180 is memorised without the picture. Drawing the triangles is what makes sense of the 2.

What helps: Redraw the diagonals from one corner. Count the triangles for a quadrilateral, a pentagon and a hexagon and notice there are always 2 fewer than the number of sides.

How to help at home

Tear and join a triangle. Draw any triangle, tear off its three corners and fit them together along a straight edge. They always make a straight line, which is 180°. Try a quadrilateral and see what total you get.
Spot parallel lines around you. Window frames, railings, a ladder or a garden gate with a diagonal bar have parallel lines crossed by another line. Find the equal pairs of angles and ask what you can say about the rest.
Walk a polygon. Walk a regular hexagon in a garden or hall: take a few steps, turn 60°, repeat six times. Your child will feel why the turns make a full 360° and why each turn is 360 ÷ 6.
Challenge with a missing angle. Draw a figure with a few angles marked and ask for the unknown one, giving a reason for every step. Swap roles: let your child set you a question and mark your reasons.

Builds on

Year 6: Units, area and volume
Year 6: Angles, circles and nets
Year 6: Four-quadrant coordinates, translation, reflection

Leads on to

Key Stage 4: Enlargement, similarity and vectors
Key Stage 4: Circles, solids and bearings
Key Stage 4: Pythagoras and trigonometry
Key Stage 4: Circle theorems and vector proof
Key Stage 4: Sine rule and cosine rule