**Circles ncert solutions Chapter 10 Exercise 10.2 Question 13**

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

**Solution:**

** Given:** ABCD is a quadrilateral and it circumscribes a circle. **Let O be the center of the circle.**

**To Prove: **

**Proof:**

In and

OP = OQ {Radius of circle}

OA = OA (Common side}

AP = AQ {Tangents drawn from an external point to the circle are equal}

Therefore, by **SSS congruence axiom**,

Therefore, {Corresponding angles of congruent triangles are equal} **(1)**

Similarly, ** (2)**

And, ** (3)**

And, ** (4)**

We also have,

Using equations** (1), (2), (3) and (4)** in the above equation, we can say that

**Hence proved**