In fig. o is the centre of a circle , prove that angle x + angle y = angle z.






Given: APB=x, AQB=y and AOB=z.To prove:   x+y=z.As shown in figure, Let ACB=c and ADB=d. QCP=(180°-c) and QDP=(180°-d)So in quadrilateral, PCQDCPD+PDQ+DQC+QCP=360°                                                   (As we know, sum of all angle of quadrilateral is 360°)x+ (180°-d)+y+(180°-c)=360°x+y=360°-360°+c+dx+y=c+d                                                           (1)We know that, AOB=2ACB                  (By the theorem, the angle formed at the center of the circle by the linesoriginating from two points on the circle's circumference is double the angle formedon the circumference of the circle by lines originating from the same points.) AOB=2ACB z=2c                  (2) AOB=2ADB z=2d                  (3)By adding (2) and (3),2c+2d=2zz=c+d                                                           (4)From equation (1) and (4), Henced proved, x+y=z  

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