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 $180^\circ$ , and a full turn is $360^\circ$ . 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 $180^\circ$ ". 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.

Open on YouTube

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 $180^\circ$ . Angles around a point add up to $360^\circ$ . Vertically opposite angles — formed where two straight lines cross — are equal.

Angles in a triangle add up to $180^\circ$ ; angles in a quadrilateral add up to $360^\circ$ . The base angles of an isosceles triangle are equal, and each angle of an equilateral triangle is $60^\circ$ .

Example 1An isosceles triangle has one angle of

40^\circ

between its two equal sides. Find the other two angles.

1The other two angles are the equal base angles.

180 - 40 = 140

, and

140 \div 2 = 70

Answer

70^\circ

and

70^\circ

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

In a quadrilateral, three of the angles are $95^\circ$ , $110^\circ$ and $72^\circ$ . Find the fourth angle, in degrees.

Your answer

Show worked solution

Solution

1Angles in a quadrilateral add up to $360^\circ$ .
2 $360 - 95 - 110 - 72 = 83^\circ$ .

Answer: $83^\circ$

Parallel lines

When a straight line crosses two parallel lines: $\textbf{alternate}$ angles (a Z shape) are equal; $\textbf{corresponding}$ angles (an F shape) are equal; $\textbf{co-interior}$ angles (a C or U shape) add up to $180^\circ$ .

a

and

b

are alternate angles, so

a = b

b

and

c

are angles on a straight line, so

b + c = 180^\circ

Example 1Two parallel lines are crossed by a transversal. One of the angles is

68^\circ

. Find its co-interior angle, giving a reason.

1Co-interior angles add up to

180^\circ

180 - 68 = 112

Answer

112^\circ

(co-interior angles add up to

180^\circ

)

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

Angle $p$ and an angle of $127^\circ$ are corresponding angles on parallel lines. What is $p$ , in degrees?

Your answer

Show worked solution

Solution

1Corresponding angles are equal, so $p = 127^\circ$ .

Answer: $127^\circ$

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 $\textbf{exterior}$ angles of any polygon add up to $360^\circ$ . At each vertex, interior angle + exterior angle $= 180^\circ$ .

The $\textbf{interior}$ angles of an $n$ -sided polygon add up to $(n - 2) \times 180^\circ$ , because the polygon can be split into $n - 2$ triangles from one vertex. In a regular polygon, all the angles are equal.

Example 1Find the size of each interior angle of a regular 12-sided polygon.

1Exterior angle

= 360 \div 12 = 30^\circ

2Interior angle

= 180 - 30 = 150^\circ

Answer

150^\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

Show worked solution

Solution

1 $(7 - 2) \times 180 = 5 \times 180 = 900^\circ$ .

Answer: $900^\circ$

Example 2Each interior angle of a regular polygon is

156^\circ

. How many sides does it have?

1Exterior angle

= 180 - 156 = 24^\circ

2Number of sides

= 360 \div 24 = 15

Answer

15

sides

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

Each exterior angle of a regular polygon is $40^\circ$ . How many sides does it have?

Your answer

Show worked solution

Solution

1 $360 \div 40 = 9$ sides.

Answer: $9$ sides

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 $360^\circ$ , 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.

Example 1In triangle

ABC

AB = AC

and angle

BAC = 2x

. Angle

ABC = x + 30

. Find

x

1Base angles of an isosceles triangle are equal: angle

ACB = x + 30

2Angles in a triangle:

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

4x + 60 = 180

, so

x = 30

Answer

x = 30

(angles

60^\circ

60^\circ

60^\circ

— the triangle is equilateral)

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

The angles of a pentagon are $x$ , $2x$ , $100^\circ$ , $110^\circ$ and $120^\circ$ . Find $x$ , in degrees.

Your answer

Show hint

Interior angles of a pentagon add up to $(5 - 2) \times 180$ .

Show worked solution

Solution

1Sum $= 540^\circ$ .
2 $3x + 330 = 540$ , so $3x = 210$ and $x = 70$ .

Answer: $x = 70$

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

\text{line } 180^\circ, \ \text{point } 360^\circ

Basic facts.

\text{alternate} = \text{corresponding}, \ \text{co-interior sum } 180^\circ

Parallel lines.

(n - 2) \times 180^\circ, \quad \text{exterior sum } 360^\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 = 60^\circ

That is the exterior angle. Interior

= 180 - 60 = 120^\circ

Co-interior angles are equal.

They add up to

180^\circ

Sum of interior angles of a hexagon

= 6 \times 180

(6 - 2) \times 180 = 720^\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 $360^\circ$ .

Reveal solution

SolutionHexagon interior angle

= 120^\circ

; square

= 90^\circ

. The gap is

360 - 120 - 90 = 150^\circ

. A regular polygon with interior angle

150^\circ

has exterior angle

30^\circ

, so

360 \div 30 = 12

sides — a regular dodecagon.

\blacksquare

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 $180^\circ$ ; point $360^\circ$ ; vertically opposite angles are equal.
Triangle $180^\circ$ ; quadrilateral $360^\circ$ .
Parallel lines: alternate and corresponding equal; co-interior sum to $180^\circ$ .
Polygon interior sum $(n - 2) \times 180^\circ$ ; exterior sum $360^\circ$ .
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 $134^\circ$ and $97^\circ$ .

Work out the third angle. Give a reason for your answer.[2]
Show mark scheme
- B1 $129^\circ$
- B1Angles around a point add up to $360^\circ$
Answer: $129^\circ$

Question 2Grade 43 marks

$ABCD$ is a parallelogram. Angle $DAB = 3x + 10$ and angle $ABC = 2x - 5$ .

Work out the size of angle $DAB$ .[3]
Show mark scheme
- M1Co-interior angles (or adjacent angles in a parallelogram) sum to $180^\circ$ : $5x + 5 = 180$
- A1 $x = 35$
- A1Angle $DAB = 115^\circ$
Answer: $115^\circ$

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]
Show mark scheme
- M1Forms $e + 7e = 180$
- M1Exterior angle $= 22.5^\circ$ , and $360 \div 22.5$
- A1 $16$ sides
Answer: $16$

Unlimited practice

Skill drill

A new question every time, marked instantly, with a full worked solution. Aim for a streak of five, then move up a level.

0 / 0 first-time correct Streak 0Best at this level: 0

Generating a question…

Test yourself

Answer a few questions, then review anything you missed.

Prefer paper? Open the written self-check

Answer each question before opening its answer. This self-check does not record a quiz score.

Two angles on a straight line are $x$ and $65^\circ$ . Find $x$ .
- $115^\circ$
- $295^\circ$
- $25^\circ$
- $65^\circ$
Show answer and explanation
Answer: $115^\circ$
$180 - 65$ .
Two angles of a triangle are $48^\circ$ and $77^\circ$ . The third is…
- $55^\circ$
- $125^\circ$
- $235^\circ$
- $65^\circ$
Show answer and explanation
Answer: $55^\circ$
$180 - 48 - 77$ .
Alternate angles are…
- equal
- add up to $180^\circ$
- add up to $90^\circ$
- add up to $360^\circ$
Show answer and explanation
Answer: equal
A Z-shape on parallel lines.
Exterior angle of a regular decagon (10 sides)?
- $36^\circ$
- $144^\circ$
- $18^\circ$
- $1440^\circ$
Show answer and explanation
Answer: $36^\circ$
$360 \div 10$ .
Sum of the interior angles of an octagon?
- $1080^\circ$
- $1440^\circ$
- $720^\circ$
- $360^\circ$
Show answer and explanation
Answer: $1080^\circ$
$6 \times 180$ .
Co-interior angles are $3x$ and $x$ . Find $x$ .
- $45^\circ$
- $60^\circ$
- $90^\circ$
- $22.5^\circ$
Show answer and explanation
Answer: $45^\circ$
$4x = 180$ .
A regular polygon has interior angles of $140^\circ$ . How many sides?
- $9$
- $8$
- $10$
- $7$
Show answer and explanation
Answer: $9$
Exterior $40^\circ$ ; $360 \div 40$ .
An isosceles triangle has a base angle of $52^\circ$ . The top angle is…
- $76^\circ$
- $64^\circ$
- $128^\circ$
- $52^\circ$
Show answer and explanation
Answer: $76^\circ$
$180 - 2 \times 52$ .

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