Answer :

Given:

POR = 130°


The inscribed angle theorem states that an angle θ inscribed in a circle is half of the central angle 2θ that subtends the same arc on the circle.


Property 1: Sum of all angles of a triangle = 180°


Property 2: Sum of all angles of a straight line = 180°


Property 3: The tangent at a point on a circle is at right angles to the radius obtained by joining center and the point of tangency.


By above property, PQO = 90°.


POR = 130°


TOR = 130°


Now,


TOR = 130°


By inscribed angle theorem,



By property 2,


TOR + TOQ = 180°


TOR + TOQ = 180°


TOQ = 180° - TOR


TOQ = 180° - 130°


TOQ = 50°


Now By property 1,


PQO + QPO + POQ = 180°


QPO = 180° - (PQO + POQ)


QPO = 180° - (90° + 50°)


QPO = 180° - 140°


1 = QPO = 40°


Hence, 1 = 40° and 2=65°


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