Question

in the given figure , o is the centre of the circle .prove that angle x = angle y + angle z .

Answer 1

see figure
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proof
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AB is a chord of the circle.  Chord AB makes an angle at the center which is twice the angle it makes at a point on the circumference.  
   hence  w = v = x/2
in the quadrilateral,  CGDH,
         angle C = y  (vertical angle)
 sum of angles in the quadrilateral CGDH is
     180 - v + y + 180 - w + z = 360
  hence,         w + v  =    x = y + z

Answer 2

Answer:

∠EBF = 1/2∠EOF = 1/2∠z [∵ Angle subtended by an arc of a circle at the centre in twice the angle subtended by it at any point of the remaining part of the circle]

[∵ Angle subtend by any arc of a circle at the centre is twice the angle subtended by it at any point of the remaining part of the circle]

In quadrilateral ABCD ∠ABC + ∠BCD + ∠CDA + ∠BAD = 2 × 1800

[ Angle Sum Property of a quadrilateral]

=> 1800 - 1/2∠z + ∠y + 1800 - 1/2∠z + ∠x = 2 × 1800

=> ∠x + ∠y = ∠z

#ananya