Question

prove that the angle subtended by an Arc at the centre is double the angle subtended by it at the remaining part of the circle

Answer 1

Given an arc PQ of a circle subtending angles POQ at the centre O and PAQ at a point A on the remaining part of the circle.
To prove= POQ = 2PAQ
PROOF
JOIN AO AND EXTENT IT TO A POINT B
NOW,
BOQ=OAQ+AQO. (exterior angle property)
In ∆OAQ ,
OA= OQ (RADIUS)
OAQ=OQA( BASE ANGLES OF ISOSCELES ∆ ARE EQUAL)
BOQ=2OAQ. ----1eq.
BOP=2OAP. ------2eq.
From 1 and 2
BOP+BOQ=2(OAP+OAQ)
OR. POQ=2PAQ
HENCE PROVED

Answer 2

Refer to NCERT book you will get your answer over there.
Thank you