Mix Examples - Circles Questions in English

(C) In a cyclic quadrilateral,the sum of opposite angles is $180^{\circ}$.
Therefore,$\angle B + \angle D = 180^{\circ}$.
Given that $\angle B = \angle D - 30^{\circ}$,we can express $\angle D$ as $\angle D = \angle B + 30^{\circ}$.
Substituting this into the sum equation: $\angle B + (\angle B + 30^{\circ}) = 180^{\circ}$.
$2\angle B + 30^{\circ} = 180^{\circ}$.
$2\angle B = 150^{\circ}$.
$\angle B = 75^{\circ}$.

(D) Let the angle subtended by the chord $AB$ at the centre be $\angle APB = \theta$.
According to the theorem,the angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle.
Therefore,$\angle APB = 2 \times \angle ACB$,which implies $\angle ACB = \frac{\theta}{2}$.
Given that $\angle ACB + \angle APB = 150^{\circ}$.
Substituting the values,we get $\frac{\theta}{2} + \theta = 150^{\circ}$.
$\frac{3\theta}{2} = 150^{\circ}$.
$\theta = 150^{\circ} \times \frac{2}{3} = 100^{\circ}$.
Thus,$\angle APB = 100^{\circ}$.