What are the circle theorems?

Angles in circles follow set rules: centre is twice the circumference, same segment angles are equal, cyclic quadrilaterals sum to 180°, and tangents meet radii at 90°. 1. The angle at the centre is twice the angle at the circumference on the same arc. 2. Angles in the same segment are equal. 3.

Properties of Shapes Notes

Q: How do you find interior and exterior angles?

Interior angles sum to (n-2) \times 180° and exterior angles to 360°. For a regular polygon, divide by n. For n sides, the interior angles sum to (n - 2) \times 180°. The exterior angles of any polygon sum to 360°.

AQA GCSE Maths: Revision notes

Key facts

Angles at a point sum to 360°; angles on a straight line sum to 180°.
Alternate and corresponding angles are equal; co-interior angles sum to 180°.
Interior angles of a polygon sum to $(n-2) \times 180°$ ; exterior angles always sum to 360°.
The angle at the centre is twice the angle at the circumference.
Opposite angles of a cyclic quadrilateral sum to 180°; a tangent meets the radius at 90°.

Angles on lines

Angles at a point sum to 360°, on a straight line to 180°, and parallel lines create equal and supplementary pairs.

Angles at a point sum to 360°. Angles on a straight line sum to 180°. Vertically opposite angles are equal.

A line crossing parallel lines is a transversal. It makes alternate angles (equal), corresponding angles (equal) and co-interior angles (sum to 180°).

Always name the fact you use.

Parallel lines and a transversal: the two 60° angles are alternate.

Alternate Corresponding Co-interior Vertically opposite Position Opposite sides, between the parallels Same side, one in and one out Same side, both between the parallels Opposite sides of a crossing Property Equal Equal Sum to 180° Equal

	Alternate	Corresponding	Co-interior	Vertically opposite
Position	Opposite sides, between the parallels	Same side, one in and one out	Same side, both between the parallels	Opposite sides of a crossing
Property	Equal	Equal	Sum to 180°	Equal

Alternate

Position:: Opposite sides, between the parallels
Property:: Equal

Corresponding

Position:: Same side, one in and one out
Property:: Equal

Co-interior

Position:: Same side, both between the parallels
Property:: Sum to 180°

Vertically opposite

Position:: Opposite sides of a crossing
Property:: Equal

Two parallel lines are crossed by a transversal. One co-interior angle is 65°. What is the other?

Triangles and quadrilaterals

Each triangle and quadrilateral has its own sides, angles, diagonals and symmetry; quote the properties that make it special.

Angles in a triangle sum to 180°; angles in any quadrilateral sum to 360°.

A trapezium has one pair of parallel sides. A kite has two pairs of adjacent equal sides, one line of symmetry and perpendicular diagonals.

To identify a shape, name the properties, not just its look.

A kite: two pairs of adjacent equal sides (a and b), and diagonals that cross at 90°.

Equilateral Isosceles Scalene Right-angled Sides 3 equal sides 2 equal sides no equal sides any sides Angles all 60° 2 equal base angles no equal angles one 90° angle Lines of symmetry 310 1 if isosceles, else 0

	Equilateral	Isosceles	Scalene	Right-angled
Sides	3 equal sides	2 equal sides	no equal sides	any sides
Angles	all 60°	2 equal base angles	no equal angles	one 90° angle
Lines of symmetry	3	1	0	1 if isosceles, else 0

Equilateral

Sides:: 3 equal sides
Angles:: all 60°
Lines of symmetry:: 3

Isosceles

Sides:: 2 equal sides
Angles:: 2 equal base angles
Lines of symmetry:: 1

Scalene

Sides:: no equal sides
Angles:: no equal angles
Lines of symmetry:: 0

Right-angled

Sides:: any sides
Angles:: one 90° angle
Lines of symmetry:: 1 if isosceles, else 0

Square Rectangle Parallelogram Rhombus Sides all equal opposite equal opposite equal and parallel all equal Angles all 90° all 90° opposite equal opposite equal Diagonals equal, bisect at 90° equal, bisect bisect each other bisect at 90° Lines of symmetry 4202

	Square	Rectangle	Parallelogram	Rhombus
Sides	all equal	opposite equal	opposite equal and parallel	all equal
Angles	all 90°	all 90°	opposite equal	opposite equal
Diagonals	equal, bisect at 90°	equal, bisect	bisect each other	bisect at 90°
Lines of symmetry	4	2	0	2

Square

Sides:: all equal
Angles:: all 90°
Diagonals:: equal, bisect at 90°
Lines of symmetry:: 4

Rectangle

Sides:: opposite equal
Angles:: all 90°
Diagonals:: equal, bisect
Lines of symmetry:: 2

Parallelogram

Sides:: opposite equal and parallel
Angles:: opposite equal
Diagonals:: bisect each other
Lines of symmetry:: 0

Rhombus

Sides:: all equal
Angles:: opposite equal
Diagonals:: bisect at 90°
Lines of symmetry:: 2

Quadrilateral	Properties
Square	All sides equal; all angles 90°; diagonals equal and bisect at 90°; 4 lines of symmetry
Rectangle	Opposite sides equal and parallel; all angles 90°; diagonals equal and bisect; 2 lines of symmetry
Parallelogram	Opposite sides equal and parallel; opposite angles equal; diagonals bisect each other (not at 90°)
Rhombus	All sides equal; opposite sides parallel; opposite angles equal; diagonals bisect at 90°; 2 lines of symmetry
Trapezium	One pair of parallel sides; no line symmetry (unless isosceles trapezium)
Kite	Two pairs of adjacent equal sides; one line of symmetry; diagonals perpendicular

Which quadrilateral has diagonals that bisect each other at 90° but are not equal in length?

Polygon angles

Interior angles sum to $(n-2) \times 180°$ and exterior angles to 360°. For a regular polygon, divide by $n$ .

For $n$ sides, the interior angles sum to $(n - 2) \times 180°$ . The exterior angles of any polygon sum to 360°.

In a regular polygon each exterior angle is $\dfrac{360°}{n}$ , and interior + exterior = 180°.

For an irregular polygon, subtract the known angles from the total to find the missing one.

A regular hexagon: each interior angle is 120°.

Interior sum $(n-2) \times 180°$
Exterior sum360°
Regular exterior $\dfrac{360°}{n}$

Interior angle sum grows by 180° with every extra side.

Worked example

Find each interior angle of a regular hexagon.

1
A hexagon has $n = 6$ sides.
2
Sum of interior angles $= (6 - 2) \times 180° = 720°$ .
3
Divide by 6: $720° \div 6 = 120°$ .

Worked example

A pentagon has interior angles of 120°, 110°, 130° and 125°. Find the fifth angle.

1
A pentagon has 5 sides: $(5 - 2) \times 180° = 540°$ .
2
Add the known angles: $120° + 110° + 130° + 125° = 485°$ .
3
Subtract: $540° - 485° = 55°$ .

A regular polygon has interior angles of 150°. How many sides does it have?

Parts of a circle

A circle is described by its radius, diameter, chord, arc, sector, segment and tangent.

The radius runs from the centre to the circumference; the diameter is twice that and passes through the centre.

A chord joins two points on the circumference. An arc is part of the circumference.

A sector is a slice between two radii and an arc. A segment lies between a chord and its arc. A tangent touches the circle at exactly one point.

Parts of a circle: radius OC, diameter AB, chord DE and a tangent touching at C.

Which part of a circle is the region between a chord and its arc?

Circle theorems

Angles in circles follow set rules: centre is twice the circumference, same segment angles are equal, cyclic quadrilaterals sum to 180°, and tangents meet radii at 90°.

The angle at the centre is twice the angle at the circumference on the same arc.
Angles in the same segment are equal.
Opposite angles of a cyclic quadrilateral sum to 180°.
A tangent is perpendicular to the radius at the point of contact.
Alternate segment theorem (Higher): the angle between a tangent and a chord equals the angle in the alternate segment.

Name the theorem in your working.

Angle at the centre (80°) is twice the angle at the circumference (40°).

A cyclic quadrilateral: opposite angles B and D sum to 95° + 85° = 180°.

The tangent at T is perpendicular to the radius OT.

Worked example

An angle at the circumference is 35°. Find the angle at the centre on the same arc.

1
Name the theorem: the angle at the centre is twice the angle at the circumference.
2
Calculate: $2 \times 35° = 70°$ .

A cyclic quadrilateral has one angle of 102°. What is the opposite angle?

Try an exam question

The angle at the circumference of a circle, standing on an arc, is 35°. Find the angle at the centre on the same arc and give a reason. A cyclic quadrilateral has one angle of 102°; find the opposite angle and give a reason.

[4 marks]

That's the notes covered.

Carry on to the next subtopic.