Question
In the given figure, if $\angle\text{AOB} = 80^\circ$ and $\angle\text{ABC} = 30^\circ,$ then find $\angle\text{CAO.}$
✓
Answer
Consider the given circle with the centre ‘O’. Let the radius of this circle be ‘r’. ‘AB’ forms a chord and it subtends an angle of 80° with its centre, that is $\angle\text{AOB}=80^\circ.$ The angle subtended by an arc at the centre of the circle is double the angle subtended by the arc in the remaining part of the circle. 
So, here we have$\angle\text{ACB}=\frac{\angle\text{AOB}}{2}$
$=\frac{80^\circ}{2}$
$\angle\text{ACB}=40^\circ$
In any triangle the sum of the interior angles need to be equal to 180°. Consider the triangle $\triangle\text{AOB}$$\angle\text{AOB}+\angle\text{OAB}+\angle\text{OBA}=180^\circ$
Since, OA = OB = r, we have $\angle\text{OAB}=\angle\text{OBA}.$ So the above equation now changes to$\angle\text{AOB}+\angle\text{OAB}+\angle\text{OBA}=180^\circ$
$2\angle\text{OAB}=180^\circ-\angle\text{AOB}$
$=180^\circ-80^\circ$
$2\angle\text{OAB}=100^\circ$
$\angle\text{OAB}=50^\circ$
Considering the triangle $\triangle\text{ABC}$ now,$\angle\text{ACB}+\angle\text{OAB}+\angle\text{OAC}+\angle\text{ABC}=180^\circ$
$\angle\text{OAC}=180^\circ-\angle\text{ACB}-\angle\text{OAB}-\angle\text{ABC}$
$=180^\circ-40^\circ-50^\circ-30^\circ$
$\angle\text{OAC}=60^\circ$
Hence, the measure of $\angle\text{CAO}$ is 60°.
So, here we have$\angle\text{ACB}=\frac{\angle\text{AOB}}{2}$$=\frac{80^\circ}{2}$
$\angle\text{ACB}=40^\circ$
In any triangle the sum of the interior angles need to be equal to 180°. Consider the triangle $\triangle\text{AOB}$$\angle\text{AOB}+\angle\text{OAB}+\angle\text{OBA}=180^\circ$
Since, OA = OB = r, we have $\angle\text{OAB}=\angle\text{OBA}.$ So the above equation now changes to$\angle\text{AOB}+\angle\text{OAB}+\angle\text{OBA}=180^\circ$
$2\angle\text{OAB}=180^\circ-\angle\text{AOB}$
$=180^\circ-80^\circ$
$2\angle\text{OAB}=100^\circ$
$\angle\text{OAB}=50^\circ$
Considering the triangle $\triangle\text{ABC}$ now,$\angle\text{ACB}+\angle\text{OAB}+\angle\text{OAC}+\angle\text{ABC}=180^\circ$
$\angle\text{OAC}=180^\circ-\angle\text{ACB}-\angle\text{OAB}-\angle\text{ABC}$
$=180^\circ-40^\circ-50^\circ-30^\circ$
$\angle\text{OAC}=60^\circ$
Hence, the measure of $\angle\text{CAO}$ is 60°.
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