Question

In the figure, O is the centre and AB is a diameter of the circle find the measure of ∠APC .Give reasons also​

Answer 1

Answer:

<apc = 30°

Step-by-step explanation:

<doc = 180-120 = 60°

<apc is the angle subtended by the arc AD

∴<apc = 1/2 <cod (The angle subtended by an arc at the center is double the angle subtended by it at any point on the remaining part of the circle.)

∴<apc = 60/2

<apc = 30°

Answer 2

$\large\underline{\sf{Solution-}}$

As AOB is a line.

So, $\rm \: \angle BOD + \angle AOD = 180\degree$

$\rm \: 120\degree + \angle AOD = 180\degree$

$\rm\implies \:\rm \: \angle AOD = 60\degree$

Now, We know,

Angle subtended at the centre of a circle by an arc is double the angle subtended on the circumference of circle by the same arc.

Now, arc AD subtends ∠APD on the circumference and ∠DOA on the center of the circle.

So, it means $\rm \: \angle AOD = 2\angle APD$

$\rm \: 2\angle APD = 60\degree$

$\rm \: \rm\implies \:\angle APD = 30\degree$

Now,

$\rm \: \angle APC\: = \:\angle APD = 30\degree$

$\rm \:\bf\implies \: \angle APC = 30\degree$

$\rule{190pt}{2pt}$

Additional Information :-

1. Angle in same segments are equal.

2. Angle in semi-circle is right angle.

3. Sum of the opposite angles of a cyclic quadrilateral is supplementary.

4. Exterior angle of a cyclic quadrilateral is equals to interior opposite angle.