How to prove that these two angles are equal? In the above figure, how to prove that $\angle$ $Oab$ and $\angle$ $OAB$ are equal in measure?
Angles $Oab$ and $Oba$ are equal as $\triangle Oab$ is isosceles. That, plus the fact that the sum of the angles in a triangle is $180^\circ$, gives you:
$$\angle Oab=\frac12(180^\circ-\angle aOb)$$
Similarly, $\angle OAB=\frac12(180^\circ-\angle AOB)$. Now just note that $\angle aOb=\angle AOB$.
Oab and OAB are similar triangles . So, the corresponding sides have same ratios, this automatically proves that these two angles are equal.