Angle Between Two Tangents

Question

In the picture below, the angle $AOB$ is $\delta \theta$, and then it is deduced that the angle between the two tangents is the same from the fact that the angles in a quadrilateral add up to $2 \pi$. However, I cannot see the quadrilateral that's being referred and so I can't possibly see how the angle between the two tangents could be $\delta \theta$.

Answer 1

The sum of two opposite angles $ = 90^0 + 90^0$. Hence there is a circle (red) passing through OAXB where X is the point of intersection of the tangents.

Answer 2

I cannot see the quadrilateral that's being referred

Let $C$ be the intersection point of the two tangents.

Then, consider the quadrilateral $OACB$ and note that $$\angle{OAC}=\angle{OBC}=\pi/2.$$