## 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 $$90^\circ$$.