# arc theroem .Please explain it

1) The angle subtended by an arc of a circle at the center is double the angle subtended by it at any point on the remaining part of the circle.
Given : Arc PQ of a circle C(O, r) and point r on the remaining part of the circle.
Prove that: ∠ POQ = 2∠PRQ
Construction: join RO and produce it to point M outside the circle.

Case 1: When arc PQ is a minor arc (figure (i))
 Statements Reasons 1) ∠POM = ∠OPM + ∠ORP 1) By exterior angle theorem (∠POM is exterior angle) 2) OP = OR 2) Radii of same circle 3) ∠OPR = ∠ORP 3) In a Δ two sides are equal then the angle opposite to them are also equal. 4) ∠POM = ∠ORP + ∠ORP 4) Substitution property. From (1) 5) ∠POM = 2∠ORP 5) Addition property.

Similarly, when ∠QOM is an exterior angle then ∠QOM = 2∠ORQ
∴ from above, ∠POQ = 2∠PRQ

Case 2: When arc PQ is a semicircle.[figure(ii)]
 Statements Reasons 1) ∠POM = ∠OPR + ∠ORP 1) Exterior angle theorem. 2) ∠POM = ∠ORP + ∠ORP 2) As,OP = OR = radius. ∴∠ORQ = ∠ORP 3) ∠POM = 2∠ORP 3) Substitution and addition property. 4) ∠QOM = ∠ORQ + ∠OQR 4) Exterior angle theorem. 5) ∠QOM = ∠ORQ + ∠ORQ 5) As, OQ =OR = radius, ∴ ∠ORQ = ∠OQR 6) ∠QOM = 2∠ORQ 6) Substitution and addition property.

• 0
What are you looking for?