Cyclic Quadrilaterals

Circle theorem: Opposite angles in a cyclic quadrilateral add up to 180°

This theorem states that if any quadrilateral is formed by four points that are on the circumference of a circle, then the angles opposite each other will add up to 180°
A cyclic quadrilateral must have all four vertices on the circumference

The theorem only works for cyclic quadrilaterals
- Do not be fooled by other quadrilaterals in a circle
- The diagram below shows a common scenario that is NOT a cyclic quadrilateral

If giving the cyclic quadrilateral theorem as a reason in an exam, use the key phrase
- ” Opposite angles in a cyclic quadrilateral add up to 180°“
- The word supplementary means angles that add up to 180° and could be used here as well you must reference a cyclic quadrilateral

Exam Tip

Identifying cyclic quadrilaterals quickly in a busy circle theorem question can help find angles and speed up answering these questions in an exam
Just remember to check that all four vertices lie on the circumference

Worked example

The circle below has centre, O, find the value of x.

This is a busy diagram with a lot going on.
Identify both the cyclic quadrilateral and the radius that is perpendicular to the chord.

Add to the diagram as you work through the problem.

The radius bisects the chord and so creates two congruent triangles.
Use this to work out 72° (equal to the equivalent angle in the other triangle) and 18° (angles in a triangle add up to 180°).

Then use the cyclic quadrilateral theorem.

2 x + 4 + 20 + 18 = 180

2 x = 138

x = 69°

1. Number

Number Toolkit

Mathematical Operations
Negative Numbers
Money Calculations
Number Operations
Related Calculations
Counting Principles

Prime Factors, HCF & LCM

Types of Number
Prime Factor Decomposition
HCF & LCM

Powers, Roots & Standard Form

Powers, Roots & Indices
Standard Form

Fractions

Basic Fractions
Operations with Fractions

Percentages

Basic Percentages
Working with Percentages

Simple & Compound Interest, Growth & Decay

Interest & Depreciation
Exponential Growth & Decay

Fractions, Decimals & Percentages

Converting between FDP
Converting between FDP

Rounding, Estimation & Bounds

Rounding & Estimation
Bounds

Surds

Simplifying Surds
Rationalising Denominators

Using a Calculator

Using a Calculator

2. Algebra

Algebra Toolkit

Algebraic Notation & Vocabulary
Algebra Basics

Algebraic Roots & Indices

Algebraic Roots & Indices

Expanding Brackets

Expanding Single Brackets
Expanding Multiple Brackets

Factorising

Factorising
Factorising Quadratics
Quadratics Factorising Methods

Completing the Square

Completing the Square

Rearranging Formulae

Rearranging Formulae

Algebraic Proof

Algebraic Proof

Linear Equations

Solving Linear Equations

Solving Quadratic Equations

Solving Quadratic Equations
Quadratic Equation Methods

Simultaneous Equationsr

Simultaneous Equations

Iteration

Iteration

Forming & Solving Equations

Forming Equations
Equations & Problem Solving

Functions

Functions Toolkit
Composite & Inverse Functions

Coordinate Geometrys

Coordinates
Coordinate Geometry

Linear Graphs y = mx + c

Straight Line Graphs (y = mx + c)
Parallel & Perpendicular Lines

Graphs of Functions

Types of Graphs
Graphical Solutions
Trig Graphs

Equation of a Circle

Equation of a Circle
Equation of a Tangents

Estimating Gradients & Areas under Graphs

Finding Gradients of Tangents
Finding Areas under Graphs

Real-Life Graphs

Distance-Time & Speed-Time Graphs
Conversion Graphs
Rates of Change of Graphs

Solving Inequalities

Solving Linear Inequalities
Conversion Graphs
Solving Quadratic Inequalities

Graphing Inequalities

Graphing Inequalities

Transformations of Graphs

Reflections of Graphs

Sequences

Introduction to Sequences
Types of Sequences
Linear Sequences
Quadratic Sequences

3. Ratio, Proportion & Rates of Change

Ratio Toolkit

Simple Ratio
Working with Proportion

Ratio Problem Solving

Ratios & FDP
Multiple Ratios

Direct & Inverse Proportions

Direct & Inverse Proportion

Standard & Compound Units

Time
Unit Conversions
Compound Measures

Exchange Rates & Best Buys

Exchange Rates & Best Buys

4. Geometry & Measures

Geometry Toolkit

Symmetry
2D & 3D Shapes
Plans & Elevations

Angles in Polygons & Parallel Lines

Basic Angle Properties
Angles in Polygons
Angles in Parallel Lines

Bearings, Scale Drawing, Constructions & Loci

Bearings
Scale & Maps
Constructing Triangles
Constructions & Loci

Circle Theorems

Angles at Centre & Semicircles
Chords & Tangents
Cyclic Quadrilaterals
Segment Theorems
Circle Theorem Proofs

Area & Perimeter

Area & Perimeter
Problem Solving with Areas

Circles, Arcs & Sectors

Area & Circumference of Circles
Arcs & Sectors

Volume & Surface Area

Volume
Surface Area

Congruence, Similarity & Geometrical Proof

Congruence
Similarity
Geometrical Proof

Area & Volume of Similar Shapes

Similar Area & Volumes

Right-Angled Triangles – Pythagoras & Trigonometry

Pythagoras Theorem
Right-Angled Trigonometry
Exact Trig Values

Sine, Cosine Rule & Area of Triangles

Sine & Cosine Rules
Area of a Triangle
Applications of Trigonomet

3D Pythagoras & Trigonometry

3D Pythagoras & Trigonometry

Vectors

Introduction to Vectors
Working with Vectors

Transformations

Translations
Reflections
Rotations
Enlargements
Combination of Transformations

5. Probability

Probability Toolkit

Basic Probability
Relative & Expected Frequency

Simple Probability Diagrams

Two Way Tables
Frequency Trees
Set Notation & Venn Diagrams

Tree Diagrams

Tree Diagrams

Combined & Conditional Probability

Combined Probability
Conditional Probability
Combined Conditional Probabilities

6. Statistics

Statistics Toolkit

Mean, Median & Mode
Averages from Tables
Range & Quartiles
Comparing Distributions
Population & Sampling

Statistical Diagrams

Bar Charts & Pictograms
Pie Charts
Time Series Graphs
Working with Statistical Diagrams

Histograms

Histograms

Cumulative Frequency & Box Plots

Cumulative Frequency Diagrams
Box Plots

Scatter Graphs & Correlation

Scatter Graphs

Top Rated by Parents/Students Nationwide