G1, G3, G4, G6Geometry & MeasuresFoundation & HigherGrades 1–5

Angles

Angle questions are chains of small facts: angles on a straight line, around a point, in a triangle, and between parallel lines. Each step is simple — the skill is choosing the right fact, and on AQA papers, writing the reason for every step.

45 min Video by Maths Genie AQA GCSE Maths

The big picture

Almost every angle fact comes from two ideas: a straight line is $18 0^{\circ}$ , and a full turn is $36 0^{\circ}$ . The triangle sum, the quadrilateral sum and the polygon formulas all follow from splitting shapes into triangles.

"Give reasons for your answer" means naming the fact in words, such as "alternate angles are equal" or "angles in a triangle add up to $18 0^{\circ}$ ". Answers without reasons often lose half the marks, even when the angle is correct.

AnglesCurated by Kepler, not produced by us. Watch it, then read the notes below.

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What you'll learn

Use angle facts on straight lines, around points and at vertically opposite angles
Use alternate, corresponding and co-interior angles in parallel lines
Use angle properties of triangles and quadrilaterals
Calculate interior and exterior angles of polygons

Basic angle facts

Angles on a straight line add up to $18 0^{\circ}$ . Angles around a point add up to $36 0^{\circ}$ . Vertically opposite angles — formed where two straight lines cross — are equal.

Angles in a triangle add up to $18 0^{\circ}$ ; angles in a quadrilateral add up to $36 0^{\circ}$ . The base angles of an isosceles triangle are equal, and each angle of an equilateral triangle is $6 0^{\circ}$ .

1The other two angles are the equal base angles.

180 - 40 = 140

, and

140 \div 2 = 70

Answer

7 0^{\circ}

and

7 0^{\circ}

Your turnSame method, new numbers — try it before moving on.

In a quadrilateral, three of the angles are $9 5^{\circ}$ , $11 0^{\circ}$ and $7 2^{\circ}$ . Find the fourth angle, in degrees.

Your answer

Parallel lines

When a straight line crosses two parallel lines: $alternate$ angles (a Z shape) are equal; $corresponding$ angles (an F shape) are equal; $co-interior$ angles (a C or U shape) add up to $18 0^{\circ}$ .

a

and

b

are alternate angles, so

a = b

b

and

c

are angles on a straight line, so

b + c = 18 0^{\circ}

1Co-interior angles add up to

18 0^{\circ}

180 - 68 = 112

Answer

11 2^{\circ}

(co-interior angles add up to

18 0^{\circ}

)

Your turnSame method, new numbers — try it before moving on.

Angle $p$ and an angle of $12 7^{\circ}$ are corresponding angles on parallel lines. What is $p$ , in degrees?

Your answer

Tip — Mark the parallel lines with arrows on any diagram you draw, and label the angles you find as you go.

Angles in polygons

The $exterior$ angles of any polygon add up to $36 0^{\circ}$ . At each vertex, interior angle + exterior angle $= 18 0^{\circ}$ .

The $interior$ angles of an $n$ -sided polygon add up to $(n - 2) \times 18 0^{\circ}$ , because the polygon can be split into $n - 2$ triangles from one vertex. In a regular polygon, all the angles are equal.

1Exterior angle

= 360 \div 12 = 3 0^{\circ}

2Interior angle

= 180 - 30 = 15 0^{\circ}

Answer

15 0^{\circ}

Your turnSame method, new numbers — try it before moving on.

Find the sum of the interior angles of a heptagon (7 sides), in degrees.

Your answer

Your turnSame method, new numbers — try it before moving on.

Each exterior angle of a regular polygon is $4 0^{\circ}$ . How many sides does it have?

Your answer

Why exterior angles total 360°

Walk around any polygon and you turn through each exterior angle once, ending up facing the way you started — one full turn. That is why exterior angles always total $36 0^{\circ}$ , whatever the number of sides.

Multi-step problems with reasons

Work towards the angle you want one fact at a time. Write each angle you find on the diagram and state the reason alongside your working.

1Base angles of an isosceles triangle are equal: angle

A C B = x + 30

2Angles in a triangle:

2 x + (x + 30) + (x + 30) = 180

4 x + 60 = 180

, so

x = 30

Answer

x = 30

(angles

6 0^{\circ}

6 0^{\circ}

6 0^{\circ}

— the triangle is equilateral)

Your turnSame method, new numbers — try it before moving on.

The angles of a pentagon are $x$ , $2 x$ , $10 0^{\circ}$ , $11 0^{\circ}$ and $12 0^{\circ}$ . Find $x$ , in degrees.

Your answer

Think like an examiner

Use the exact wording for reasons: "alternate angles are equal", not "Z angles".
A diagram "not drawn accurately" means you must calculate — measuring with a protractor scores nothing.
In algebraic angle questions, find $x$ and then the angle the question asks for.

Angle facts

line 18 0^{\circ}, point 36 0^{\circ}

Basic facts.

alternate = corresponding, co-interior sum 18 0^{\circ}

Parallel lines.

(n - 2) \times 18 0^{\circ}, exterior sum 36 0^{\circ}

Polygons.

Watch out for these

Using "Z angles" or "F angles" as a written reason.

Write "alternate angles" or "corresponding angles".

Interior angle of a regular hexagon

= 360 \div 6 = 6 0^{\circ}

That is the exterior angle. Interior

= 180 - 60 = 12 0^{\circ}

Co-interior angles are equal.

They add up to

18 0^{\circ}

Sum of interior angles of a hexagon

= 6 \times 180

(6 - 2) \times 180 = 72 0^{\circ}

Stretch yourself

A regular hexagon and a square share a side, as part of a tiling pattern. A regular polygon fits exactly into the gap at the shared vertex. How many sides does that polygon have?

Hint — Angles at the point must add up to $36 0^{\circ}$ .

Questions students ask

How do I remember alternate versus corresponding?

Alternate angles sit on alternate (opposite) sides of the transversal, inside the parallel lines — a Z. Corresponding angles are in the same position at each crossing — an F. Just do not write "Z" or "F" as a reason.

Does

(n - 2) \times 180

work for irregular polygons?

Yes, the interior angle sum works for every polygon (without holes or crossing sides). Only dividing by

n

to get one angle needs the polygon to be regular.

What does "not drawn accurately" mean?

That the diagram is a sketch. Angles and lengths must be calculated from the given information, not measured.

Key takeaways

Straight line $18 0^{\circ}$ ; point $36 0^{\circ}$ ; vertically opposite angles are equal.
Triangle $18 0^{\circ}$ ; quadrilateral $36 0^{\circ}$ .
Parallel lines: alternate and corresponding equal; co-interior sum to $18 0^{\circ}$ .
Polygon interior sum $(n - 2) \times 18 0^{\circ}$ ; exterior sum $36 0^{\circ}$ .
State a reason, in words, for every step.

Exam-style questions

Practise it the way it is examined

3 original questions · 8 marks, written to match the style and mark allocation of the real papers. Work on paper, then open the mark scheme and tick the marks you earned — method marks count even if the final answer slips.

Question 1Grade 22 marks

Three angles meet at a point. Two of them are $13 4^{\circ}$ and $9 7^{\circ}$ .

Work out the third angle. Give a reason for your answer.[2]

Question 2Grade 43 marks

$A B C D$ is a parallelogram. Angle $D A B = 3 x + 10$ and angle $A B C = 2 x - 5$ .

Work out the size of angle $D A B$ .[3]

Question 3Grade 53 marks

The interior angle of a regular polygon is 7 times the size of its exterior angle. How many sides does the polygon have?[3]

Unlimited practice

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Angles

Basic angle facts

Parallel lines

Angles in polygons

Multi-step problems with reasons

Practise it the way it is examined

Skill drill

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Angles

Basic angle facts

Parallel lines

Angles in polygons

Multi-step problems with reasons

Practise it the way it is examined

Skill drill

Practise this topic

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