Is that possible that a inscribe angle can be greater than 90 degree

Question

I have found a question like following:

Its asked that what could be the angle $x$ if $BC$ is not diameter of the circle.

So, my question is if it possible to be greater then $90^{\circ}$ for an angle like $x$?

My proof said:

The highest inscribed angle only could be made of with the chord same diameter. The other chords could not create an angle greater than that.

Answer 1

Yes, the angle can get very close to (but not exactly equal to) 180 degrees. For every angle value greater than 90-degrees the chord $\overline{BC}$ is smaller than the diameter.

Answer 2

Note that you can also put a point $A'$ on the other side of the chord (i.e. on the longer arc of the circle), which would give an acute angle.

Using the theorem that the angle at the centre is twice the angle at the circumference (which you may know) you will find the angles $A+A'= 180^{\circ}$ - because the total angle at the centre is $360^{\circ}$. So if the chord is not a diameter, one of the angles will be obtuse and the other acute.