Angle properties of circles

The angle at the centre is twice the angle at the circumference, angles in the same segment are equal, and the angle in a semicircle is a right angle.

Three theorems

Angle in a semicircle

Slide P along the top of the circle. The angle subtended by the diameter is always a right angle.

Example 1●●

The angle at the centre is $110^\circ$. Find the angle at the circumference on the same arc.

$110 \div 2 = 55^\circ$ (angle at the centre is twice the angle at the circumference)

Example 2●●

$AB$ is a diameter and $C$ is on the circle with $\angle CAB = 35^\circ$. Find $\angle ABC$.

$\angle ACB = 90^\circ$ (angle in a semicircle)

$\angle ABC = 180 - 90 - 35 = 55^\circ$ (angle sum of triangle)

Flashcards

Centre vs circumference

Centre angle $= 2 \times$ circumference angle

Angle in a semicircle

$90^\circ$

Same segment

Equal angles

Practice

Angle at the centre $110^\circ$. Angle at the circumference on the same arc?
$AB$ is a diameter, $\angle CAB = 35^\circ$. Find $\angle ABC$.
Angle at the circumference $35^\circ$. Angle at the centre on the same arc?
Two angles stand on the same arc in the same segment. One is $42^\circ$. The other?
Angle subtended at the circumference by a diameter?
The angle at the circumference is $100^\circ$ (on the major arc side, a reflex centre angle). Centre angle?
$O$ is the centre and $OA = OB$ with $\angle AOB = 130^\circ$. Find $\angle OAB$.
Same circle: $C$ on the major arc. Find $\angle ACB$.