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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.
Answer :

Given: MN and PO are two equal chords of two congruent circles with centre at O and O'.

To prove: \(\angle{MON}\) and \(\angle{PO'Q}\)
image
Proof: In \(\triangle{MON}\) and \(\triangle{PO'Q}\), we have,
MO = PO' ...(Radii of congruent circles)
NO = QO' ...(Radii of congruent circles)
and MN = PQ ...(Given)
By SSS criterion, we get,
\(\triangle{MON}\) \(\displaystyle \cong\) \(\triangle{PO'Q}\)
Hence, \(\angle{MON}\) = \(\angle{PO'Q}\) ...(By CPCT)