The words you need
Circle theorems come with their own vocabulary, and half the battle is reading the diagram in those terms.
- Chord
- A straight line joining two points on the circle. A chord splits the circle into two segments.
- Arc
- Part of the circle's edge between two points. Angles are described as "standing on" the arc between their end points.
- Tangent
- A straight line that touches the circle at exactly one point without crossing into it.
Angles at the centre
Two of the most-used theorems involve the centre \(O\) of the circle:
- Angle at the centre: the angle at the centre is twice the angle at the circumference when both stand on the same arc.
- Angle in a semicircle: an angle drawn at the circumference on a diameter is always \(90^\circ\). It is just the special case where the "arc" is a half-circle, so the centre angle is \(180^\circ\) and half of that is \(90^\circ\).
Segments and quadrilaterals
Two more theorems compare angles at the circumference:
- Angles in the same segment: angles at the circumference standing on the same arc are equal. Move the vertex anywhere along the same segment and the angle does not change.
- Cyclic quadrilateral: if all four corners of a quadrilateral lie on the circle, its opposite angles add to \(180^\circ\).
Always name the theorem
In your working, write the reason next to each step, such as "angles in the same segment". The reasoning earns marks even when the arithmetic is short.
Tangent theorems
The last two involve tangents:
- Tangent and radius: a tangent meets the radius at the point of contact at exactly \(90^\circ\).
- Alternate segment theorem: the angle between a tangent and a chord equals the angle in the alternate segment (the angle at the circumference on the other side of that chord).
| Theorem | What it tells you |
|---|---|
| Angle at the centre | Centre angle \(= 2 \times\) circumference angle |
| Angle in a semicircle | Angle on a diameter \(= 90^\circ\) |
| Same segment | Angles on the same arc are equal |
| Cyclic quadrilateral | Opposite angles sum to \(180^\circ\) |
| Tangent and radius | Meet at \(90^\circ\) |
| Alternate segment | Tangent-chord angle \(=\) angle in alternate segment |
A find-the-angle example
Most questions chain two or three theorems together. Take them one step at a time, writing the reason each time.
Points \(A\), \(B\) and \(C\) lie on a circle with centre \(O\). The angle at the centre \(\angle AOC = 84^\circ\). Find the angle \(\angle ABC\) at the circumference, standing on the same arc \(AC\).
Where this is assessed
Recalling the correct theorem is Criterion A, and setting out a clear chain of reasoning with a stated reason at every step is Criterion C. A right answer with no reasons rarely gets full marks.
Check yourself
1. A triangle is drawn inside a circle with one side as a diameter. What is the angle opposite the diameter? +
By the angle-in-a-semicircle theorem it is exactly \(\mathbf{90^\circ}\).
2. In a cyclic quadrilateral one angle is \(110^\circ\). What is the angle opposite it? +
Opposite angles of a cyclic quadrilateral sum to \(180^\circ\), so the opposite angle is \(180^\circ - 110^\circ = \mathbf{70^\circ}\).
3. The angle at the centre standing on an arc is \(100^\circ\). What is the angle at the circumference on the same arc? +
The circumference angle is half the centre angle: \(\tfrac{1}{2}\times 100^\circ = \mathbf{50^\circ}\).
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