How to find $\angle{EAD}$ in the given figure? 
(It is not a homework problem! It was asked in one of my friends' placement related entrance examination this year.)
My work: Assuming $\angle{EAD}=\angle{\mathbf{1}}=x,$ it is clear that $\angle{ODB}=x$ since $\angle{AEB}=90^\circ,$ being a semi-circular angle.
(Now if I consider $BC$ to be a tangent to the circle at $B,$ then using all the known facts we can find $\angle{ACB}$ as well as $\angle{DAO}.$ But still I am not able to find $x.$)
Although it is not mentioned that $BC$ is a tangent so considering $BC$ to be a secant, I am not able to make use of the fact that $OB=BC.$ (I think I should not consider $BC$ as a tangent.) Either way I am stuck basically.
Please give some hints to make use of the fact: $OB=BC$ for finding $x.$ Thanks in advance.
 A: 
I am assuming $BC$ is tangent to the circle as you mentioned in your work.
We have $\triangle ADO \sim \triangle ABC$
Also, $\triangle ADO \sim \triangle AFB$
Using similarity, what is the relation between $BF$ and $FD$?
Finally note that $x = \angle FBD$
A: Assuming $CB$ tangent at $B$ (see @Ivan Kaznacheyeu comment), then join $CO$.
Since $BCDO$ is a cyclic quadrilateral, and right triangle $COB$ is isosceles, then$$\angle COB=45^o$$But $$\angle CDB=\angle COB$$since they stand on common arc $CB$.
Hence$$\angle EDA=\angle CDB=45^o$$Therefore right triangle $ADE$ is also isosceles and$$\angle EAD=45^o$$
A: I respectfully disagree with Math Lover, Edward, and OP's assumption of tangency of $BC$ to the large circle. Below, I first explain the construction, then I show with an example that the problem indeed lacks sufficient information, then I give a general guideline for solving the problem. I will conclude with remarks about the problem.
1.Construction
Reproducing the construction helps us better understand the problem. We can start from the large circle with center $O$ and radius $OA$. Let's call it the main circle.

*

*Within the main circle, draw $Circle1$ with diameter $OA$.

*From $B$ draw a line such that it intersects (or touches) $Circle1$. Let this line first meet $Circle1$ at $D$ and then meet the main circle at $E$. Already we have constructed the angles $\widehat{AEB}=\widehat{ODA}=90^o$ (why?) and the unknown angle $\widehat{EAD}$, the measure of which we wish to find.

*Draw $Circle2$ with center $B$ and radius $OB$.

*Extend $AD$ to meet (or touch) $Circle2$. Name the furthest meeting point of this extension as $C$.


2.Information
The above figure shows two examples of the construction. In the lower example, $BC$ is tangent to the main circle. Note that

*

*$\widehat{EAD}$ measures differently in the two examples. This means that the problem in its current wording does not have a unique answer. Comparing the measures with the multiple choices of the problem shows that the designers of the problem had assumed that $BC$ is tangent to the main circle, but they did not mention this assumption in the problem. Therefore, the problem lacks information necessary for the intended solution.

*The construction of the angle $\widehat{EAD}$ was completed without any regard to point $C$. In the problem as it is worded, especially without mentioning the constraint of tangency of $BC$ to the main circle, the information given about $BC$ and $C$ is useless.

3.General guideline
The interested reader can solve the problem using analytic geometry, and find the measure of $\widehat{EAD}$ based on $\widehat{EBA}$, which defines the slope of the line $BE$, which was arbitrarily drawn in step 2 of the construction (considering the only constraint that it should meet $Circle1$). The general steps are:

*

*Write the equations of the main circle, $Circle1$ and $BE$

*Find $D$, the intersection of $BE$ and $Circle1$.

*Find $E$, the intersection of $BE$ and the main circle.

*Calculate the angle between $AD$ and $AE$.

4.Conclusion
Designers of the problem failed to mention all of the assumptions that they had when designing the problem. Without the particular assumption of tangency of $BC$ to the main circle, the question has an infinite number of answers. Moreover, without this assumption, the information given about point $C$ is useless. I was only able to figure out the assumptions of the designers from the multiple choices of the question. This is an example of a bad question that wastes the time of everyone who works on it. Therefore, while I respect the efforts made by the OP, I give a down vote to the question itself. I hope my act encourages OP and other MSE members to create for themselves some sort of a blacklist of sources that are not worthy of our time and effort.
