THEOREM 1

The angle which an arc of a circle subtends at the centre of a circle is twice that which it subtends at any point on the circumference of the circle.

Given: A circle PQR, centre O.
To prove that P Ô Q = 2 P R Q
Construction: Join RO, and extend to any point T.

Proof: Po =Ro= Qo (radii)
a = a1 (base ∴ P Ô Q = a + a1 = 2a (end But P Ô T + Q Ô T
Reflex of P Ô Q = 2a + 2b = 2(a +b) = 2 PRQ
∴ angle at centre = 2 x (at circum. Of a circle).

Corollary

The angle is a semi circle is a right angle
Given: A semi circle3 P Q R, Centre O.