Question

the angle subtended by an Arc at the centre is double the angle subtended by it at any point on the remaining part of circle​

Answer 1

Given : An arc ABC of a circle with center O , and a point C on the remaining part of the circumference.

To Prove : Angle AOB =  Twice angle ABC

Construction : Join OC and produce it to a suitable point D

Proof : Let angles 1,2,3,4,5,6 be as shown in figure

             a) OA=OB=OC [RADII OF THE SAME CIRCLE ARE EQUAL]

             b)1=4,2=3 [ANGLES OPPOSITE TO EQUAL SIDES ARE EQUAL]

             c)5=4+1=Twice angle1 [EXTERIOR ANGLE PROPERTY]

             d)6=2+3=Twice angle 2 [EXTERIOR ANGLE PROPERTY]

             e)5+6=Twice angle (1+2) [ADD STATEMENT c AND d]

             f)Angle AOB = Twice angle ACB [WHOLE = SUM OF ALL ITS PARTS]