Angles at Centre & Circumference (Edexcel IGCSE Maths)

Spec point — 4.6C

Angles at centre & circumference

What are circle theorems?

• Circle Theorems deal with angles that occur when lines are drawn within (and connected to) a circle

• You may need to use other facts and rules such as:

  • basic properties of lines and angles
  • properties of triangles and quadrilaterals
  • angles in parallel lines or polygons

Parts of a circle

Circle Theorem: The angle at the centre is twice the angle at the circumference

• In this theorem, the chords (radii) to the centre and the chords to the circumference are both drawn from (subtended by) the ends of the same arc

• To spot this circle theorem on a diagram

  • find any two radii in the circle and follow them to the circumference
  • see if there are lines from those points going to any other point on the circumference
  • it may look like the shape of an arrowhead

• When explaining this theorem in an exam you must use the keywords:

  • The angle at the centre is twice the angle at the circumference

• This theorem is still true when the ‘triangle parts’ overlap

• It is also true when the lines form a diamond shape

  • You need to compare the reflex angle at the centre with the angle at the circumference

  • Common mistakes are to

    • compare the wrong angles
    • confuse it with a different circle theorem on cyclic quadrilaterals

Exam Hint Questions often say to give “reasons” for your answer.

Quote an angle fact or circle theorem for every angle you find (not just one for the final answer).

Worked Example Find the value of $x$* *in the diagram below.

Not drawn accurately

Give a reason for each step of your working.

Answer:

*There are three radii in the diagram, AO, BO and *CO Mark these as equal length lines

Notice how they create two isosceles triangles Base angles in isosceles triangles are equal

Angle* OAB = angle OBA = 60º (isosceles triangle)*

Use the circle theorem:

The angle at the centre is twice the angle at the circumference

Form an equation for $x$

$2 ( x + 60 ) = 150$

Expand the brackets and solve the equation

$\begin{array}{rcl} 2 x + 120 & = 150 \\ 2 x & = 30 \\ x & = 15 \end{array}$

$x=15$

Final answer: Base angles in isosceles triangles are equal

Final answer: The angle at the centre is twice the angle at the circumference