Question

If $ABCD$ is a cyclic quadrilateral in which $AD$ is the diameter and $\angle BCD = 125^\circ $, then the measure of $\angle ADB$ is
A. $35^\circ $
B. $45^\circ $
C. $55^\circ $
D. $65^\circ $

Answer 1

Hint: Here we are asked to find the angle $\angle ADB$ by using the given data. First, we will try to make a diagram out of given data. A cyclic quadrilateral is a quadrilateral which is drawn inside a circle by touching the circle at its vertices $A,B,C$ and $D$. And by using the property of a cyclic quadrilateral that is the sum of the opposite angles equals $180^\circ $ we will find the required angle.

Complete step-by-step answer:
It is given that $ABCD$ is a cyclic quadrilateral in which $AD$ is the diameter and angle $\angle BCD = 125^\circ $ we aim to find the angle $\angle ADB$ so let us first draw a diagram for a better view of this problem.

By the property of a cyclic quadrilateral, we know that the sum of the opposite angle of a cyclic quadrilateral equals $180^\circ $ so we get $\angle BCD + \angle BAD = 180^\circ $.
We are given that angle $\angle BCD = 125^\circ $ substituting this we get
$ \Rightarrow 125^\circ + \angle BAD = 180^\circ $
To find the angle $\angle BAD$ let us take the term $125^\circ $ to the other side.
$\angle BAD = 180^\circ - 125^\circ $
$ \Rightarrow \angle BAD = 55^\circ $
Now we got the angle $\angle BAD$ but we need to find the angle $\angle ADB$ .
Now let us draw the diagonal $BD$ to the quadrilateral.

From the diagram we can see that the angle $\angle ABD$ is in a semi-circle thus it will be equal to $90^\circ $.
We already know that $\angle BAD = 55^\circ $ now we know that $\angle ABD = 90^\circ $ so in the triangle $\Delta ABD$ in the diagram we have $\angle A = 55^\circ $ and $\angle B = 90^\circ $ so the third angle $\angle D$ is
$\angle D = 180^\circ - \angle A - \angle B$
$ \Rightarrow \angle D = 180^\circ - 55^\circ - 90^\circ $
$ \Rightarrow \angle D = 35^\circ $
From this we can say that the angle $\angle ADB = 35^\circ $ .
Thus, option a) $35^\circ $ is the correct option.

So, the correct answer is “Option A”.

Note: In the above problem, we have used the property of a triangle to find the required angle that is the sum of all the three angles in a triangle will always be equal to $180^\circ $ with is we can write $\angle BAD + \angle ADB + \angle DBA = 180^\circ $ for the triangle $\Delta ABD$. Thus, to find the angle $\angle ADB$ we rewritten the above as $\angle ADB = 180^\circ - \angle BAD - \angle DBA$.