Solve problems involving angles in circles

Angles at the Circumference and Central Angle Subtended by an Arc

6.1

Angles at the Circumference and Central Angle Subtended by an Arc

Angles at the circumference of a circle:

	Example

	(i) The diagram below shows two chords, \(PQ\) and \(QR\) which meet at point \(Q\) at the circumference of the circle.

	\(\angle PQR \) is the angle at the circumference of the circle subtended by the arc \(PR\).

	(ii)

	\(\angle PQS\) and \(\angle PRS\) are angles at the circumference of the circle subtended by the major arc \(PS\). \(\angle QPR\) and \(\angle QSR\) are angles at the circumference of the circle subtended by the minor arc \(QR\).

Rules of angles in a circle:

Angles at the circumference subtended by the same arc are equal.
Angles at the circumference subtended by arcs of the same length are equal.
The size of an angle at the circumference subtended by an arc is proportional to the arc length.
The size of the angle at the centre of a circle (central angle) subtended by the same arc is twice the size of the angle at the circumference.

	Example

	The following diagram shows a circle with a length of arcs \(PR=QS\).

	Determine the value of \(x\).

	The value of \(x=40 {^\circ}\). This is because \(\angle x\) and \(\angle 40{^\circ}\) are at the circumference and the length of arcs \(PR=QS\).

	Example

	The following diagram shows a circle.

	Determine the value of \(x\) and \(y\) .

	The value of \(x \) is \(35^\circ\). Meanwhile, the value of \(y\) is \(\begin{aligned} y&= 35{^\circ}(2) \\\\y&= 70{^\circ}. \end{aligned}\)

Central angles of a circle:

For central angles of a circle subtended by an arc:

The sizes of the angles are equal if their arc lengths are equal.
The change in the size of an angle is proportional to the change in the arc length.

Value of angles at the circumference subtended by the diameter:

The angle at the circumference of a circle subtended by the diameter is \(90^\circ\).