Question

angle PQR = 100°, where P, Q and Rare points on a circle with centre O. Find angel OPR.




please anyone can solve it​

Answer 1

the measure of angle OPR is 40

Answer 2

given: In circle with centre O

∆PQR, angle PQR = 100°

to find: angle OPR = ?

construction: subtend angle PSR in the major segment

solution:

quadrilateral PQRS is cyclic

hence,

angle PQR + angle PSR = 180°

... (opposite angles of cyclic quadrilateral are supplementary)

100° + angle PSR = 180°

angle PSR = 80°

angle POR = 2 angle PSR

... (angle subtended by an arc at the centre of the circle is double the angle subtemded by it on any point on the circle)

angle POR = 2 x 80°

angle POR = 160° ... (i)

now, in ∆POR,

OP = PR ... (radii of same circle)

hence,

angle OPR = angle ORP ... (ii)

... (angles opposite to equal sides of a triangle are equal)

angle OPR + angle ORP + angle POR = 180°

angle OPR + angle OPR + 160° = 180° (from i and ii)

2angle OPR = 20°

angle OPR = 10°