What is the value of x?

If we look at figure angle o is 2y

both the angle of the sides which pass through o is z

the 2 equations formed are
2z+2y=180
z+y=90

x+75+(90-z)=180
x-z=15

using both options individually, we can find the answer , so answer is D

What is the value of x?
A) Y = 55 deg
B) Z = 35 deg

#1
central angle = 110 , but no other info ; insufficient
#2
z= 35 , other angle not know ; insufficient
from 1&2
No new info
IMO E

Ans:D
The center angle = 2y.
hence, 2y+z+z =180 (For the triangle at center of circle)--------eqn (i)
Let the angle opposite to 75 degree be N (for the bottom most triangle)
Hence, N+Z=90(triangle at center of circle is isosceles) ----------eqn(ii)
75+N+x=180---------------------------------------------------------eqn(iii)
From eqn (i , ii and iii)
75+y+x=180---------------------------------------------------------eqn(iv)
(1)Y is given hence x can be found from eqn iv (Sufficient)
(2)Z is given hence x can be found from eqn ii and eqn iii(Sufficient)

Lets us first arrive at the equations basis given information y degree is marked as vertex D and side consecutive to D on left side is AD and on right side is BD and for x degree vertex is marked as C. Now Angle AOB = 2 y degree( By virtue of property that central angle is twice the subtended angle which is y degree or angle ADB).
OA=OB so angle OAB = angle OBA = z degree, angle BAC = 90-z as entire angle A is 90 degrees and angle OAB = z degrees
In triangle OAB , 2y+z+z = 180 so y+z=90 degrees....... eq 1
and in triangle BAC 90-z+75+x = 180 so x-z= 15 degrees...... eq 2
A) Y = 55 deg
so from here we can arrive at value of x after combining equation 1 and equation 2, x = 50 so statement 1 alone is sufficient
B) Z = 35 deg
From eq 2 x comes to 50 degrees so statement 2 alone is also sufficient
Hence Answer is Option D

Refer the attached sheet for explanation.

Imo. D

The answer is E.

EXPLANATION:

Let ABC be the triangle inscribed in the circle. And BXZ be the triangle inscribes outside the circle.

Now, We are not given, whether O is the center of the circle or not.
So we can conclusively decide. Even if both statements are given, we won't be able to find X.

NOTE:
If O is the center of the circle, then, D would be the answer.

This is because, OB and OC would be same (Radius), making OBC triangle as isoceles.

Using statement 1:
Y = 55 degrees. The angle subtended at the center would be twice. So Angle BOC = 110 degrees.
So angle Z and and angle OBC = 35 degrees (as 110 + x + x = 180, means 2x = 70, and x = 35).

So the angle XBZ = 90 - 35 = 55 degrees. So X = 180 - (55 + 75) = 45 degrees.

Using Statement 2:
We are directly given Z = 35 degrees. Following the above logic, we would arrive at the same result.

Hence, if O is the center of the circle, answer is D, else, the answer is E, as we are not given whether O is the center or not.

The answer should D
st 1: y=55 so angle at center will be 2*55=110 using subtended angle property.
since both the side connected to angle O are radius , hence angle opposite to these sides can be found by using triangle property.
therefore, angle z= (180-110)/2= 35
as angle opposite 75 is given as 90 Deg (combined with upper and lower triangle), we can get the angle in the lower triangle as we know angle of the upper part
so angle opposite 75 will be -- 90-35=65
we have two angles 65 & 75 we can get the x= 180-75-65= 40
so statement is sufficient

st2 : value of angle z =35 is given , following the above steps , we can get the angle of x-- sufficient

OA=D

In the given figure, if O is the center of the circle, then the question can be proceeded as follows:
By alternate segment theorem, v=y
(1) if y= 55 deg, v = y = 55 deg. Therefore, x = 180 - (75+y) = 180 - 130 = 50 deg. Clearly, sufficient.
(2) if z =35 deg, v= y = 90 - 35 = 55 deg. Therefore, x = 180 - (75+y) = 180 - 130 = 50 deg. Clearly, sufficient.
Therefore, both the statements are sufficient.

But, O is not given as the center of the circle. Choice E is the answer.

In the above figure, since it is not mentioned that whether O is the center of the circle or not, nothing can be commented about the value of x.
(1) No unique value.
(2) No unique value.
(1) + (2) No unique value.

E is the answer.

Thats a tricky one. The question never mentioned O is the centre of the circle. That makes the answer E.

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Yes, You are right anurasi , as we cannot assume anything on our own in DS questions. We have to just solve it by logic + Data Given.

IMO-E