# Prove that if chords of congruent circles subtend equal angles at their centres, then the chords are equal.

Given:

Chords of congruent circles subtend equal angles at their centres.

To do:

We have to prove that the chords are equal.

Solution:

Let $c_{1}$ and $C_{2}$ be two congruent circles, $AB$ and $PQ$ are their chords respectively.

Let us join $OA$ and $OB$ in circle $C_{1}$.

Similarly, in cirlcle $C_{2}$, join $MP$ and $MQ$.

In $\vartriangle OAB$ and $\vartriangle MPQ$.

$OA=MP$ [$\because$ Radius of the congruent circles are same]

$OB=MQ$ [$\because$ Radius of congruent circles are same]

$\angle AOB=\angle PMQ$ [It is given chords of congruent circles subtend equal angles at their centres]

$\Rightarrow \vartriangle OAB\cong \vartriangle MPQ$ [SAS rule of congruency]

$\therefore AB=PQ$ [By CPCT rule]

Hence, it has been proved that the chords are equal.

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