Question

o is a centre of the circle prove that X + Y is equal to Z

Answer 1

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, PCQD

∠CPD+∠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+d

x+y=c+d                                                          

 (1)

We know that, ∠AOB=2∠ACB

                  (By the theorem, the angle

 formed at the center of the circle by the

 lines

originating from two points on the circle's

 circumference is double the angle formed

on the circumference of the circle by lines

 originating from the same points.)

∵ ∠AOB=2∠ACB

⇒ z=2c                  (2)

∵ ∠AOB=2∠ADB

⇒ z=2d                  (3)

By adding (2) and (3),

2c+2d=2z

⇒z=c+d                                                          

 (4)

From equation (1) and (4),

 

Henced proved, ∠x+∠y=∠z