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In the given figure, O is the centre of the circle, prove that ∠x = ∠y + ∠z. - Mathematics

Short Note

In the given figure, is the centre of the circle, prove that ∠x = ∠y + ∠z. 

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Solution

It is given that, O is the center of circle and AB and C are points on circumference on triangle

We have to prove that x = ∠y + ∠z

∠4
 and ∠3 are on same segment

So, ∠4 = ∠3           
    `angle x = 2 angle3`  (Angle subtended by an arc of a circle at the centre is double the angle subtended by it at any point on the remaining part of the circle)

 `anglex = angle 4 + angle 3`               …… (1)

 `angley= angle 3 + angle 1`           (Exterior angle is equal to the sum of two opposite interior angles)         …… (2)

\[\angle4 = \angle z + \angle1\]

   (Exterior angle is equal to the sum of two opposite interior angles)  

  `anglez= angle 4 -  angle 1`              …… (3)

Adding (2) and (3)

  `angley + angle z  = angle 3 + angle 4`     ……(4)

From equation (1) and (4) we have 

 `anglex = angle y + angle z`     

 

  Is there an error in this question or solution?
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APPEARS IN

RD Sharma Mathematics for Class 9
Chapter 15 Circles
Exercise 15.4 | Q 10 | Page 74
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