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Circle : Exercise 10.2 (Mathematics NCERT Class 9th)

Q.1     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.
Sol. Given : AB and  CD are two equal  chords of a circle  with center at O.
To prove : $\angle AOB = \angle COD$

Proof : In $\Delta s$ AOB and COD,  we have
AO = CO                       [Radii of the same circle]
BO = DO                       [Radii of the same circle]
and     AB = CD            [Given]
Therefore  by  SSS criterion  of congruence, we have
$\Delta AOB \cong \Delta COD$
$\Rightarrow$    $\angle AOB = \angle COD$                   [C.P.C.T.]

Q.2      Prove that if chords of congruent circles subtend  equal  angles at their  centres, then the chords are equal.
Sol.

Given : AB and CD are two chords such that angles subtended by these chords at the centre of the  circle are equal. i.e.          $\angle AOB = \angle COD$
To prove :         AB = CD
Proof  : In $\Delta s$ AOB and  COD we have
AO = CO         [Radii of the same  circle]
BO = DO         [Radii of the same circle]
and $\angle AOB = \angle COD$
Therefore   By SAS criterion  of congruence,  we have
$\Delta \,AOB \cong \Delta COD$
$\Rightarrow$        AB = CD                         [C.P.C.T.]

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