# Ex.10.2 Q1 Circles Solution - NCERT Maths Class 9

## Question

Recall that two circles are congruent if they have the same radii. Prove that equal chords of congruent circles subtend equal angles at their centers.

## Text Solution

**What is known?**

Two circles are congruent if they have same radii.

**What is unknown?**

Equal chords of congruent circles subtend equal angles at their centers.

**Reasoning:**

Using chords are equal and the fact that circles are congruent, we prove the statement using Side-Side-Side (SSS criteria) and Corresponding parts of congruent triangles (CPCT).

**Steps:**

Let \(\begin{align} {QR} \end{align}\) and \(\begin{align} {YZ} \end{align}\) be the equal chords of **\(2\)** congruent circles.

\(\begin{align} {QR = YZ} \end{align}\)

We need to prove that they subtend equal angles at centre. i.e. \(\begin{align}\angle{QPR} =\angle {YXZ}\end{align}\)

We know that the radii of both the circles are equal. So we get:

\(\begin{align} {PR = PQ = XZ = XY} \end{align}\)

Consider the **\(2\)** triangles \(\begin{align} \Delta { PQR} \end{align}\) and \(\begin{align} \Delta { XYZ} \end{align}\).

\(\begin{align} &{PQ = XY} \qquad \text{(Radii are equal)}\\&{PR = XZ} \qquad \text{(Radii are equal)}\\&{QR = YZ} \qquad \text{(Chords are equal)}\end{align}\)

By **SSS criteria**, \(\begin{align} \Delta\,{PQR}\end{align}\) is congruent to \(\begin{align} \Delta\;{XYZ}.\end{align}\)

So by **CPCT** (Corresponding parts of congruent triangles) we get \(\begin{align} \angle{QPR}=\angle{YXZ}.\end{align}\)

Hence proved that equal chords of congruent circles subtend equal angles at their centres.