# Circle Theorem: Chord of a Circle, Theorem 1, Theorem 2, Theorem 3, Theorem 4, Theorem 5 (For CBSE, ICSE, IAS, NET, NRA 2022)

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## Circle Theorem: Theorem 4, Theorem 5

(4) To any point on the circumference of the circle, the measure of the angles subtended from the same arc is equal to half of the angle subtended at the center by the same arc.

From the above given figure,

Join passing through

In

Since angles opposite to equal sides of a triangle are equal,

By exterior Angle Property of the triangle, we have

Thus, from equation (ii) and (iii) , we have

Similarly, in ,

Exterior angle,

From equation (iv) and (v) , we have

, Hence Proved

(5) The opposite angles in a cyclic quadrilateral are supplementary.

Proof:

Let us consider arc .

using theorem (4)

For arc ,

Reflex using theorem (4)

From (i) and (ii) , we get

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