From figure,
Consider PR as a chord of the circle.
Steps of construction :
Mark any point S on the major arc of the circle. (on the side opposite to point Q)
Join PS and SR.
PQRS is a cyclic quadrilateral.
⇒ ∠PQR + ∠PSR = 180° (Sum of opposite angles in a cyclic quadrilateral = 180°)
⇒ 100° + ∠PSR = 180°
⇒ ∠PSR = 180° - 100° = 80°.
We know that,
The angle subtended by an arc at the center is double the angle subtended by it at any point on the remaining part of the circle.
∴ ∠POR = 2∠PSR
⇒ ∠POR = 2 × 80°
⇒ ∠POR = 160°.
In ∆ OPR,
⇒ OP = OR (Radius of the circle)
We know that,
Angles opposite to equal sides are equal.
∴ ∠OPR = ∠ORP = a (let).
We know that,
Sum of all angles in a triangle is 180°.
⇒ ∠OPR + ∠POR + ∠ORP = 180°
⇒ a + 160° + a = 180° [∵ ∠OPR = ∠ORP]
⇒ 2a = 180° - 160°
⇒ 2a = 20°
⇒ a = = 10°
⇒ ∠OPR = 10°.
Hence, ∠OPR = 10°.