Angle properties of circles
Explore the angle properties of circles for Year 11 Specialist Mathematics in Queensland (QCAA). These theorems connect the angle at the centre, the angle at the circumference, the right angle in a semicircle, and the angles of a cyclic quadrilateral.
You will learn to identify the arc each angle stands on, choose the right property, and find unknown angles with a stated reason — the proof foundation for the tangent, chord and geometric proofs later in Unit 2.
Theory
Angle properties of circles are the theorems that link angles at the centre, angles at the circumference and the angles of a cyclic quadrilateral in Year 11 Specialist Mathematics (QCAA, Queensland). This page states each property, shows the diagrams, and works through finding unknown angles step by step.
A point lies on a circle if it is on the curve itself; the centre \(O\) is the fixed point equidistant from every such point. Every line from the centre to the circle is a radius, so any two radii are equal in length.
An arc is part of the circle between two points \(A\) and \(C\). An angle stands on (is subtended by) that arc when its two arms pass through \(A\) and \(C\). The angle at the centre \(\angle AOC\) has its vertex at \(O\); an angle at the circumference (an inscribed angle) \(\angle ABC\) has its vertex \(B\) on the circle.
A cyclic quadrilateral is a four-sided figure whose four vertices all lie on one circle. A semicircle is half a circle, cut off by a diameter (a chord through the centre).
Every property below relates angles that stand on the same arc, so the first step in any problem is to identify which arc each angle rests on.
For an inscribed angle and the central angle standing on the same arc \(AC\):
Special case — the diameter gives a central angle of \(180^\circ\), so the angle in a semicircle is a right angle:
Angles in the same segment (standing on the same arc) are equal:
Opposite angles of a cyclic quadrilateral, and the exterior angle:
How to find an unknown circle angle
- Identify the arc: mark the two points the unknown angle stands on, and check which other angles (centre, circumference, or opposite vertex) stand on the same arc.
- Choose the property: centre vs circumference (halve or double), angle in a semicircle (\(90^\circ\)), same segment (equal), or cyclic quadrilateral (supplementary / exterior angle).
- Write the relationship as an equation, then substitute the known angle.
- Solve and state the reason — every answer in a proof must quote the property used.
Both angles stand on arc \(AC\), so the inscribed angle is half the central angle:
| \(\angle AOC\) | \(=\) | \(2\,\angle ABC\) |
| \(140^\circ\) | \(=\) | \(2\,\angle ABC\) |
| \(\angle ABC\) | \(=\) | \(\dfrac{140^\circ}{2}\) |
| \(=\) | \(70^\circ\) |
\(\angle ABC = 70^\circ\).
The angle in a semicircle is a right angle:
| \(\angle ACB\) | \(=\) | \(90^\circ\) |
Then use the angle sum of \(\triangle ABC\):
| \(\angle ABC\) | \(=\) | \(180^\circ - 90^\circ - \angle CAB\) |
| \(=\) | \(180^\circ - 90^\circ - 28^\circ\) | |
| \(=\) | \(62^\circ\) |
\(\angle ABC = 62^\circ\).
Angles in the same segment stand on the same arc \(AB\), so they are equal:
| \(\angle AQB\) | \(=\) | \(\angle APB\) |
| \(=\) | \(39^\circ\) |
The central angle is twice an inscribed angle on the same arc:
| \(\angle AOB\) | \(=\) | \(2\,\angle APB\) |
| \(=\) | \(2 \times 39^\circ\) | |
| \(=\) | \(78^\circ\) |
\(\angle AQB = 39^\circ\) and \(\angle AOB = 78^\circ\).
Opposite angles of a cyclic quadrilateral are supplementary:
| \(\angle DAB + \angle BCD\) | \(=\) | \(180^\circ\) |
| \(116^\circ + \angle BCD\) | \(=\) | \(180^\circ\) |
| \(\angle BCD\) | \(=\) | \(180^\circ - 116^\circ\) |
| \(=\) | \(64^\circ\) |
The exterior angle equals the interior opposite angle \(\angle DAB\) (and is supplementary to \(\angle BCD\)):
| \(\angle DCF\) | \(=\) | \(180^\circ - \angle BCD\) |
| \(=\) | \(180^\circ - 64^\circ\) | |
| \(=\) | \(116^\circ\) |
\(\angle BCD = 64^\circ\) and \(\angle DCF = 116^\circ\).
Common pitfalls
Frequently asked questions
What is the angle at the centre theorem?
The angle a chord subtends at the centre is twice the angle it subtends at the circumference, provided both stand on the same arc: \(\angle AOC = 2\,\angle ABC\).
Why is the angle in a semicircle 90 degrees?
The diameter makes a straight angle of \(180^\circ\) at the centre. The inscribed angle standing on the same arc is half of that, so it is \(90^\circ\).
What does 'angles in the same segment' mean?
Two inscribed angles that stand on the same arc, on the same side of the chord, are equal. They subtend the same arc, so \(\angle APB = \angle AQB\).
What is the rule for a cyclic quadrilateral?
The four vertices lie on a circle, and each pair of opposite angles adds to \(180^\circ\) (they are supplementary). The exterior angle equals the interior opposite angle.
How do I know which property to use?
Identify the arc the unknown angle stands on, then look for a centre angle, a diameter, another circumference angle on that arc, or an opposite vertex of a cyclic quadrilateral.
Do these properties need to be proved in Specialist Mathematics?
Yes. QCAA Unit 2 Topic 3 requires you to prove each circle property and then use it to find unknown angles and prove further results.