Angle Properties of Circles - Theorems 1 and 2
Every question with a fully worked solution.
- Angle Properties of Circles - Theorems 1 and 2 - Video - Circle Theorems - Angle at the Centre Watch
Theory
An angle at the centre of a circle is twice the angle at the circumference standing on the same arc (Theorem 1), and angles at the circumference standing on the same arc are equal (Theorem 2). This Year 10A maths page shows how to use these two theorems, including reflex angles, algebra and short proofs, with reasons at every step.
An angle at the centre has its vertex at the centre \(O\). An angle at the circumference has its vertex on the circle. Both stand on (are subtended by) the arc between the ends of their arms.
To find the arc an angle stands on, follow its two arms to the circle: the arc between those points on the far side from the vertex is the one. An angle at the circumference that stands on the major arc is obtuse, and its matching angle at the centre is a reflex angle.
Angles at the circumference that stand on the same arc are said to be in the same segment: their vertices all lie on the arc on the other side of the chord.
Theorem 1: the angle at the centre is twice the angle at the circumference standing on the same arc. Theorem 2: angles at the circumference standing on the same arc are equal. Theorem 2 follows from Theorem 1, because each angle is half of the same angle at the centre.
Every answer needs a reason, written as a short theorem name such as "angle at the centre is twice the angle at the circumference" or "angles on the same arc".
It prepares you for senior maths courses, where you prove results as well as use them.
The two theorems, for angles standing on the same arc \(AB\):
| Theorem | Reason to write |
|---|---|
| 1. Centre angle \(= 2 \times\) circumference angle | angle at the centre is twice the angle at the circumference |
| 2. Angles on the same arc are equal | angles standing on the same arc (angles in the same segment) |
| Theorem 1 with the point on the minor arc | reflex angle at the centre is twice the angle at the circumference |
How to find an angle in a circle
- Mark the given angles and any equal radii on the diagram.
- Trace the arms of each angle to find the arc it stands on. Angles on the same arc are linked: equal at the circumference, double at the centre.
- Look for an isosceles triangle with two radii, a triangle whose angle sum you can use, or a quadrilateral with \(O\) as one vertex.
- Calculate one angle at a time and write the reason beside each step.
Problem types
- Expressions in a pronumeral: use a theorem to write an equation, e.g. centre angle \(= 2\times\) circumference angle, or two angles on the same arc are equal. Solve, then substitute back to find the angle (Example 3).
- Reflex angles: an obtuse angle at the circumference pairs with the reflex angle at the centre. Subtract from \(360^\circ\) to get the non-reflex angle (Example 1).
- Chords that cross: find an angle on the same arc first, then use the angle sum of a triangle or the exterior angle of a triangle (Example 2).
- Radii and the angle at the centre: in \(\triangle OAB\) the base angles are equal, so the angle at \(O\) is \(180^\circ\) minus twice a base angle. Halve it for the angle at the circumference (Example 4).
- Quadrilateral with the centre as a vertex (e.g. \(AOBC\)): let the circumference angle be \(x\), write the reflex angle at \(O\) as \(360^\circ - 2x\), and use the angle sum of \(360^\circ\).
- Proofs: join the centre to the points on the circle, mark equal radii, name the base angles \(\alpha\), \(\beta\), and use the exterior angle of a triangle or an angle sum. To prove Theorem 1, produce \(CO\) to a point \(P\) and use the exterior angles at \(O\) of the two isosceles triangles. To prove Theorem 2, write each angle at the circumference in terms of the same angle at the centre.
(i) the non-reflex \(\angle AOC\)
(ii) \(\angle ADC\).
\(\angle ABC\) stands on the major arc \(AC\), so the angle at the centre on that arc is the reflex angle \(AOC\).
(i) Angle at the centre is twice the angle at the circumference:
| \(\text{reflex }\angle AOC\) | \(=\) | \(2 \times 118^\circ = 236^\circ\) |
| \(\angle AOC\) | \(=\) | \(360^\circ - 236^\circ\) |
| \(\) | \(=\) | \(\mathbf{124^\circ}\) |
(ii) \(\angle ADC\) stands on the minor arc \(AC\), so it is half of the non-reflex angle at the centre:
| \(\angle ADC\) | \(=\) | \(\tfrac{1}{2} \times 124^\circ = \mathbf{62^\circ}\) |
Notice that \(118^\circ + 62^\circ = 180^\circ\). This is no accident: it is Theorem 4, about cyclic quadrilaterals, which you meet next.
\(\angle BDC\) and \(\angle BAC\) both stand on arc \(BC\), so they are equal (angles on the same arc):
| \(\angle BAX\) | \(=\) | \(\angle BDC = 44^\circ\) |
In \(\triangle ABX\) (angle sum of a triangle):
| \(\angle AXB\) | \(=\) | \(180^\circ - 44^\circ - 30^\circ\) |
| \(\) | \(=\) | \(\mathbf{106^\circ}\) |
Name each angle with three letters: the middle letter is the vertex, and the outer letters show the arc it stands on.
Both angles stand on the minor arc \(AB\). The angle at the centre is twice the angle at the circumference:
| \(4m + 6\) | \(=\) | \(2(m + 18)\) |
| \(4m + 6\) | \(=\) | \(2m + 36\) |
| \(2m\) | \(=\) | \(30\) |
| \(m\) | \(=\) | \(\mathbf{15}\) |
Substitute back:
| \(\angle AOB\) | \(=\) | \(4(15) + 6 = \mathbf{66^\circ}\) |
| \(\angle ACB\) | \(=\) | \(15 + 18 = \mathbf{33^\circ}\) |
Check: \(66^\circ = 2 \times 33^\circ\) \(\checkmark\)
Let \(\angle ACB = \theta\).
| \(\angle AOB\) | \(=\) | \(2\theta\) (angle at the centre is twice the angle at the circumference) |
\(OA = OB\) (radii), so \(\triangle OAB\) is isosceles and \(\angle OAB = \angle OBA\). Angle sum of \(\triangle OAB\):
| \(2\angle OAB + 2\theta\) | \(=\) | \(180^\circ\) |
| \(\angle OAB + \theta\) | \(=\) | \(90^\circ\) |
\(\therefore \angle OAB + \angle ACB = 90^\circ\). Each step has a reason, and the last line is exactly the statement to be proved.
Common pitfalls
Frequently asked questions
What is the angle at the centre theorem?
The angle at the centre of a circle is twice the angle at the circumference when both stand on the same arc.
What does angles in the same segment mean?
Angles at the circumference that stand on the same arc have their vertices in the same segment, on the other side of the chord. These angles are equal.
What if the point on the circle is on the minor arc?
Then the angle at the circumference is obtuse, and it is half of the reflex angle at the centre. Subtract the reflex angle from 360 degrees to get the non-reflex angle.
How do you know which arc an angle stands on?
Follow the two arms of the angle to the circle. The arc between those two points, on the far side from the vertex, is the arc the angle stands on.
Do I need to give reasons in circle geometry?
Yes. Each step should name the theorem or fact used, such as angles on the same arc, radii, or angle sum of a triangle. Marks are given for the reasons as well as the answer.