Question 1197237
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P is the centre of a circle that passes through O, and O is the center of a circle that passes through P. If C = 66 degrees, then the measure of OPB is?
Please find the diagram in the link below:
https://ibb.co/1XKHRrr
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            The solution in the post by  @math_tutor_2020  is incorrect.

            I came to bring a correct solution.



<pre>
Triangle PCB is is isosceles triangle. Therefore, angle PBC = angle PCB = 66°.

Hence, angle BPC is 180° - 66° - 66° = 48°.



Let x be the measure of the angle A. 
Triangle APO is isosceles triangle, so angle APO = x.

Angle POB is an exterior angle of the triangle APO; therefore, angle POB = x+x = 2x.



Triangle POB is isosceles triangle; therefore, angle PBO = angle POB = 2x.



Write the sum of interior angles of triangle POB

    angle POB + angle PBO + angle OPB = 180°,

or

    2x + 2x + angle OPB = 180°.



It implies  angle OPB = 180° - 4x.



Next, write the sum of angles APO, OPB and BPC 

    x + (180° - 4x) + 48° = 180°.



From this equation,

    48° = 3x,

    x = 48°/3 = 16°.


<U>ANSWER</U>.  The measure of the angle OPB is 180° - 4x = 180° - 4*16° = 180° - 64° = 116°.
</pre>

Solved.