SOLUTION: 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

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Question 1197237: 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
Found 2 solutions by math_tutor2020, ikleyn:
Answer by math_tutor2020(3817)   (Show Source): You can put this solution on YOUR website!

Ignore this solution. I misread the initial set up.

Answer: 132 degrees

Reason:
Angle PCB = 66 is an inscribed angle of circle P
Use the inscribed angle theorem to find that minor arc OB is 2*66 = 132 degrees. This is the shortest measure from point O to point B along the edge of the circle.
By definition, central angle OPB is the same measure of this minor arc because the central angle subtends the arc.

If it might help, erase circle O to avoid clutter.
Answer by ikleyn(52844)   (Show Source): You can put this solution on YOUR website!
.
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
~~~~~~~~~~~~~~~


            The solution in the post by  @math_tutor_2020  is incorrect.
            I came to bring a correct solution. 


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°.


ANSWER.  The measure of the angle OPB is 180° - 4x = 180° - 4*16° = 180° - 64° = 116°.

Solved.



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