Prove that if the angles subtended by the chords of a circle (or of congruent circles) at the centre (or the corresponding centres) are equal,then the chords are equal.
(N/A) Given: In a circle with centre $O$,chords $AB$ and $PQ$ subtend equal angles at the centre,i.e.,$\angle AOB = \angle POQ$.
To Prove: $AB = PQ$.
Proof: In $\Delta AOB$ and $\Delta POQ$:
$OA = OP$ (Radii of the same circle)
$OB = OQ$ (Radii of the same circle)
$\angle AOB = \angle POQ$ (Given)
Therefore,$\Delta AOB \cong \Delta POQ$ ($SAS$ congruence rule).
Therefore,$AB = PQ$ $(CPCT)$.
To Prove: $AB = PQ$.
Proof: In $\Delta AOB$ and $\Delta POQ$:
$OA = OP$ (Radii of the same circle)
$OB = OQ$ (Radii of the same circle)
$\angle AOB = \angle POQ$ (Given)
Therefore,$\Delta AOB \cong \Delta POQ$ ($SAS$ congruence rule).
Therefore,$AB = PQ$ $(CPCT)$.
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