I am looking at a picture from a physics problem. On the right, we have $\delta \theta / 2$ for the angle between the tangent at the center of the small arc and the chord between the middle and lower(upper) point. I am not aware of a geometric result that guarantees this. Is this a mere approximation or an actual result?
In the diagram, $x$ indicates the angle $\frac{\delta\theta}{2}$ in your original diagram.
The obtuse angle opposite the $x$ between the two tangents is $180^o-x$ due to the fact that a quadrilateral has total internal angle $360^o$ and the other two angles between the tangents and the radii are $90^o$.
Then, since a straight line is $180^o$, the external angle indicated is also $x$.
I hope this helps.