How to prove that these two angles are equal?

Question

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In the above figure, how to prove that $\angle$ $Oab$ and $\angle$ $OAB$ are equal in measure?

Answer 1

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 $\angle aOb=\angle AOB$.