# CBSE Class 10-Mathematics: Questions and Answers Chapter – 10 Circles Part 20 (For CBSE, ICSE, IAS, NET, NRA 2023)

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**Question 3**:

Prove that opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the center of the circle.

**Answer**:

Given: ABCD is a quadrilateral circumscribing a circle whose center is O.

To prove: (i)

(ii)

Construction: Join OP, OQ, OR and OS.

Proof: Since tangents from an external point to a circle are equal.

In OBP and ,

[Radii of the same circle]

[Common]

[From eq. (i) ]

[By SSS congruence criterion]

Similarly,

Since the sum of all the angles round a point is equal to .

Similarly, we can prove that

**Question 4**:

In the given figure XY and X ‘Y’ are two parallel tangents to a circle with center O and another tangent AB with point of contact C intersecting XY at A and X ‘Y’ at B. Prove that .

**Answer**:

Join

In and , we have

[radius]

Similarly,

Now,

But in ,