In a circle with centre O, ACBO is a parallelogram where C is a point on the minor arc AB. What is the measure of AOB?

As per the given in the question,

Let $$\angle ACB=y, \angle AOB=x$$ and $$\angle AOB=z$$ external angle

We know that angle subtended by an arc at the center is twice the angle subtended at the circumference.

Hence, arc ADB subtend twice angle as ACB is subtending,

So $$2y=z----------------(i)$$

ACBO is a parallelogram,

so $$\angle ACB=\angle AOB$$  ------------(opposite angle of any parallelogram)

So, $$x=y-----------(ii)$$

We know that

$$\Rightarrow x+z=360$$

From equation (i) and (ii)

$$\Rightarrow y+2y=360$$

$$\Rightarrow 3y=360$$

$$\Rightarrow y=120$$