Find the missing angle x.
Angles on a straight line add to 180°.
180 − 120 = 60°.
Angles
Angles are formed when two lines meet. Understanding angle rules is essential for solving problems involving shapes, geometry and algebra. This topic links directly to trigonometry and polygons.
Overview
Angles measure how much a line turns.
In GCSE Maths, you need to know the main angle facts and use them to calculate missing angles.
Straight line = 180° | Full turn = 360°
Most angle questions follow the same pattern: identify the rule, form an equation, and solve.
What you should understand after this topic
- Recognise common angle types
- Use angle facts on lines and around a point
- Work with vertically opposite angles
- Solve parallel line problems
- Apply polygon angle rules
Key Definitions
Acute Angle
An angle less than \(90^\circ\).
Right Angle
An angle exactly \(90^\circ\).
Obtuse Angle
An angle greater than \(90^\circ\) but less than \(180^\circ\).
Straight Angle
An angle of \(180^\circ\).
Reflex Angle
An angle greater than \(180^\circ\) but less than \(360^\circ\).
Vertically Opposite Angles
Opposite angles formed by two intersecting lines. They are equal.
Interior Angle
An angle inside a shape.
Exterior Angle
An angle outside a shape, formed by extending a side.
Key Rules
Angles on a straight line
They add up to \(180^\circ\).
Angles around a point
They add up to \(360^\circ\).
Vertically opposite angles
They are equal.
Angles in a triangle
They add up to \(180^\circ\).
Angles in a quadrilateral
They add up to \(360^\circ\).
Corresponding angles
They are equal on parallel lines.
Alternate angles
They are equal on parallel lines.
Co-interior angles
They add up to \(180^\circ\).
Important Polygon Rules
Sum of interior angles
\((n - 2) \times 180^\circ\)
Sum of exterior angles
\(360^\circ\)
Regular polygon exterior angle
\(\frac{360^\circ}{n}\)
Regular polygon interior angle
\(180^\circ - \text{exterior angle}\)
How to Solve
Step 1: Recognise the type of angle fact
Before calculating anything, decide which angle rule applies. This is the most important step in solving angle problems.
Angles on a line add to \(180^\circ\).
Angles around a point add to \(360^\circ\).
Vertically opposite angles are equal.
Exam tip: Identify the rule before doing any calculation.
Step 2: Write the correct total and form an equation
Once you recognise the diagram, write the total and form an equation to find the missing angle.
Line \(=180^\circ\), Point \(=360^\circ\), Triangle \(=180^\circ\), Quadrilateral \(=360^\circ\)
You will often solve this using solving linear equations.
Why this matters: Most angle questions reduce to solving an equation.
Parallel line angles
When a transversal crosses parallel lines, several angle relationships appear together.
Corresponding angles
Equal angles in matching positions.
Alternate angles
Equal angles forming a Z-shape.
Co-interior angles
Add to \(180^\circ\) and form a C-shape.
Exam method
State the rule first, then form an equation.
Angles in triangles and quadrilaterals
Basic shapes have fixed angle totals that are used in almost every exam.
Triangles are covered in more detail in triangles.
Triangle
Sum = \(180^\circ\)
Quadrilateral
Sum = \(360^\circ\)
Polygon angles
For shapes with more than four sides, use a formula to find the total of the interior angles.
Sum of interior angles: \( (n - 2)\times180^\circ \)
This is often combined with algebra in exam questions.
Regular polygons
In a regular polygon, all sides and angles are equal.
Exterior angle \(= \frac{360^\circ}{n}\)
Interior angle \(= 180^\circ - \text{exterior angle}\)
Exam tip: Find the exterior angle first, then the interior angle.
Example Questions
Edexcel
Exam-style questions focusing on basic angle facts, straight lines and angles around a point.
Edexcel
Angles on a straight line are shown.
Find the value of x.
Edexcel
Angles around a point are shown.
Find the value of x.
Edexcel
Two straight lines intersect.
Find the value of x.
AQA
Exam-style questions focusing on triangles, isosceles triangles and parallel line angle facts.
AQA
The angles in a triangle are shown.
Find the value of x.
AQA
An isosceles triangle has two equal sides.
Find the value of x.
AQA
Two parallel lines are crossed by a transversal.
Find the value of x. Give a reason for your answer.
OCR
Exam-style questions focusing on polygon angle reasoning and regular polygons.
OCR
A nonagon is shown.
Find the sum of the interior angles.
OCR
A regular decagon has 10 equal exterior angles.
Find the size of each exterior angle.
OCR
A regular polygon has exterior angle 24°.
Find the number of sides.
Exam Checklist
Step 1
Identify the angle fact from the diagram.
Step 2
Write the correct total: \(180^\circ\), \(360^\circ\) or a polygon rule.
Step 3
Build an equation carefully.
Step 4
Check that your answer makes sense in the diagram.
Most common exam mistakes
Wrong rule
Using straight-line angles when the question is about parallel lines or polygons.
Wrong total
Using \(360^\circ\) instead of \(180^\circ\), or the other way round.
Parallel lines confusion
Mixing up corresponding, alternate and co-interior angles.
Polygon formula errors
Forgetting that interior sum uses \((n-2)\times180\).
Common Mistakes
These are common mistakes students make when solving angle problems in GCSE Maths.
Using the wrong angle rule
Incorrect
A student applies a rule that does not match the diagram.
Correct
Always identify the type of angles first (e.g. alternate, corresponding, vertically opposite) before choosing a rule. Using the wrong rule leads to incorrect answers.
Forgetting vertically opposite angles are equal
Incorrect
A student treats vertically opposite angles as different values.
Correct
Vertically opposite angles are always equal. These angles are directly opposite each other when two lines cross.
Mixing up corresponding and co-interior angles
Incorrect
A student says co-interior angles are equal.
Correct
Corresponding angles are equal, but co-interior angles add up to \(180^\circ\). It is important to recognise which pair you are working with.
Using the wrong total for angles
Incorrect
A student uses \(360^\circ\) when the angles should sum to \(180^\circ\).
Correct
Angles on a straight line add to \(180^\circ\), while angles around a point add to \(360^\circ\). Choose the correct total based on the diagram.
Forgetting how to handle regular polygons
Incorrect
A student tries to find interior angles directly without using the correct method.
Correct
For regular polygons, it is often easier to find the exterior angle first using \(360^\circ \div n\), then subtract from \(180^\circ\) to find the interior angle.
Try It Yourself
Practise the main angle facts first, then move to parallel lines and polygons.
Foundation
Foundation Practice
Use basic angle facts including straight lines, triangles and around a point.
What is the total angle around a point?
Full turn.
Angles around a point add to 360°.
Find the missing angle x.
Angles in a triangle = 180°.
180 − 50 − 60 = 70°.
Which angle is a right angle?
Quarter turn.
A right angle is 90°.
Find x.
Angles around a point = 360°.
360 − 150 − 110 = 100°.
Angles in a triangle always add up to:
Triangle rule.
Angles in a triangle = 180°.
Find x.
Straight line = 180°.
180 − 70 = 110°.
An acute angle is:
Small angle.
Acute angles are less than 90°.
Angles around a point are 120°, 130° and x. Find x.
Add to 360°.
360 − 120 − 130 = 110°.
A straight line angle is:
Half turn.
A straight line = 180°.
Higher
Higher Practice
Solve angle problems involving parallel lines and polygons.
Find x.
Alternate angles are equal.
x = 60° (alternate angles).
Corresponding angles are:
Parallel lines rule.
Corresponding angles are equal.
Find x.
Co-interior angles add to 180°.
180 − 120 = 60°.
Find the sum of interior angles of a pentagon.
Use (n − 2) × 180.
(5 − 2) × 180 = 540°.
The exterior angles of any polygon add up to:
Full turn.
Exterior angles always add to 360°.
A regular hexagon has equal angles. Find each interior angle.
Total ÷ number of sides.
720 ÷ 6 = 120°.
Alternate angles are formed when:
Think parallel lines.
Alternate angles come from a transversal crossing parallel lines.
Find x if interior angle of a regular polygon is 135°.
Work backwards using formula.
It is an octagon (8 sides).
Co-interior angles sum to:
Same side interior.
Co-interior angles add to 180°.
Find x.
Triangle angles = 180°.
180 − 70 − 70 = 40°.
Games
Practise this topic with interactive games.
Games coming soon.
Frequently Asked Questions
What is the sum of angles in a triangle?
180 degrees.
What are vertically opposite angles?
Angles opposite each other that are equal.
What are corresponding angles?
Angles in matching positions on parallel lines.