Angle Properties of a Circle
Geometry · Angle Properties of a Circle
Syllabus tag: KCSE | Mathematics | Form 2 | Topic 19 Angle Properties of a Circle
Lesson objectives
By the end of this topic, you should be able to:
- Identify an arc, chord and segment of a circle.
- Relate and compute the angle subtended by an arc at the circumference.
- Relate the angle at the centre to the angle at the circumference.
- Apply the properties of a cyclic quadrilateral.
Angle Properties of a Circle
Angles drawn inside a circle obey a small set of rules. Nearly every circle question uses one of them.
a) The parts
An arc is part of the circumference. The shorter one is the minor arc, the longer the major arc.
A chord is a straight line joining two points on the circle. The longest chord is the diameter.
A segment is the region between a chord and its arc.
An angle is subtended by an arc when lines from the arc's ends meet at a point. The angle formed there is the one subtended.
b) Angle at the centre
The angle at the centre is twice the angle at the circumference. Both must stand on the same arc.
So an angle of 40° at the circumference gives 80° at the centre, on the same arc.
Check that both angles stand on the same arc. If they do not, the rule does not apply. That check is where most marks are lost.
c) Angles in the same segment
Angles in the same segment are equal.
Any number of angles drawn from the same arc, on the same side, are all equal.
This follows from the previous rule. Each equals half the same central angle, so they must all be equal.
d) Angle in a semicircle
The angle in a semicircle is a right angle.
This is the same rule again. The diameter subtends 180° at the centre. Half of that is 90°.
Whenever a question shows a triangle with the diameter as one side, look for the right angle.
e) Cyclic quadrilaterals
A cyclic quadrilateral has all four vertices on the circle.
Opposite angles of a cyclic quadrilateral add to 180°.
Also, an exterior angle equals the interior opposite angle.
Both follow from the centre rule applied to the two arcs.
f) Chord properties
A perpendicular from the centre to a chord bisects the chord.
Equal chords are equidistant from the centre, and chords equidistant from the centre are equal.
These give a right-angled triangle from centre, midpoint and endpoint, so Pythagoras applies.
g) Working a problem
Mark every angle you know on the diagram.
Identify which arc each angle stands on. That decides which rule applies.
State the rule you are using before writing the calculation. Marks are given for the reason, not only the number.
h) Where this is used
Structural design with arches. Surveying by angles. Navigation fixes. Any geometry involving circular parts.
Words to know
- Subtend -- to form an angle at a point, by lines drawn from the ends of an arc or chord.
- Cyclic quadrilateral -- a quadrilateral with all four vertices on a circle.
- Segment -- the region between a chord and an arc.
- Major arc -- the longer of the two arcs cut off by a chord.
- Supplementary -- adding to 180°.
:::checkpoint Check yourself
- An angle at the circumference is 35°. What is the angle at the centre on the same arc?
- Why is the angle in a semicircle always 90°?
- Three angles of a cyclic quadrilateral are 85°, 95° and 100°. Find the fourth.
- What does a perpendicular from the centre do to a chord? :::
Bridge to practice
The exercises begin with subtending and the central result, move through angles in the same segment and the semicircle, and finish with cyclic quadrilaterals and chord properties. For every angle you write down, write its reason beside it before moving on.