The angles substended by a chord of a circle at any two points on the circumference of the circle which lie on one side of the chord are equal.

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The angles substended by a chord of a circle at any two points on the circumference of the circle which lie on one side of the chord are equal.

Proof:

Let us consider a circle with C as centre. Let line segment AB be a chord of the circle.

Let P and Q be any two points on the circumference of the circle on one side of the chord AB. 

Then the angle substended by the chord AB at the centre of the circle is (angle ACB).

Also,

The angles substended by the chord AB at the points P and Q are (angle APB) and (angle AQB) respectively.

Before going to the next step of this proof, let us learn this :

The angle substended by a chord of a circle at the centre of the circle is equal to double the angle substended by that chord at any point on the circumference of the circle lying on a side of the chord where centre lie.

Therefore, from above information,

(angle ACB) = 2 × (angle APB) and  

(angle ACB) = 2 × (angle AQB)

Therefore,

[2 × (angle APB)] = [2 × (angle AQB)]

i.e, (angle APB) = (angle AQB)

As P and Q are any two points and as (angle APB) = (angle AQB) , the angles substended by a chord of a circle at any two points on the circumference of the circle which lie on one side of the chord are equal. 


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