Is it possible to find $\angle BAO$ given two additional pieces of information?

Question

Figure 1 with Answer Choices

This is a GMAT practice question that I cannot solve. The correct answer states that the angle can be known with either piece of additional information, but I cannot see how both pieces of information would not be necessary to know the angle. What theorem, property, etc. allows either piece of additional info to be used to solve for $\angle BAO$?

Sorry for any errors with my post, as this is my first (which is why I am unable to embed the image of the figure and answer choices).

Answer 1

Given that : $OB=OC=AB$

let $\angle A=x$

$\angle BOA=\angle A = x$ (isosceles $\triangle$)

$\angle BOA=2x$ (exterior $\angle$ of $\triangle$)

$\angle OCB = \angle BOA=2x$ (isosceles $\triangle$)

First case $\angle COD =60^{\circ}$

$\angle COD = 3x$ (exterior $\angle$ of $\triangle$)

$\angle BOA = x=20^{\circ}$

Second case

$\angle BCO = 2x =40^{\circ}$

$\angle BOA = x=20^{\circ}$

in both cases $\angle BOA$ can be found.