Question

The measure of all angles subtended by an arc on the same segment of the circle are -

• A. equal
• B. unequal
• C. double the measure of the angle subtended at the centre by the same arc.
• D. Insufficient data.

Answer 1

Answer: (A), (C)

double the measure of the angle subtended at the centre by the same arc.

Consider the major segment CB of the following figure:

Here, angles $\angle D$ and $\angle A$ are subtended by an arc on the same segment of the circle.

We know that the angle subtended by an arc at the centre of a circle is double the angle subtended by the arc at any point on the circle.

Considering the arc CB and point D, $\angle BOC=2\times \angle BDC.$
$\Rightarrow \angle BDC=\frac{\angle BOC}{2}$

Considering the arc CB and point A, $\angle BOC=2\times \angle BAC.$
$\Rightarrow \angle BAC=\frac{\angle BOC}{2}$
$\therefore \angle BDC=\angle BAC$
This is true for any point lying on the major arc CB.

Hence, both option (a) and (c) are correct.