Question
Equal chords of a circle subtend equal angles at the centre.
✓
Answer
Proof : You are given two equal chords $A B$ and $C D$ of a circle with centre O (see Fig.9.4). You want to prove that $\angle \mathrm{AOB}=\angle \mathrm{COD}$.
In triangles $A O B$ and $C O D$,
$\mathrm{OA}$ $=\mathrm{OC}$ (Radii of a circle)
$\mathrm{OB}$ $=\mathrm{OD}$ (Radii of a circle)
$\mathrm{AB}$ $=\mathrm{CD}$ (Given)
Therefore, $\quad \triangle \mathrm{AOB}$ $\cong \Delta \mathrm{COD}$ (SSS rule)
This gives $\quad \angle \mathrm{AOB}$ $=\angle \mathrm{COD}$
(Corresponding parts of congruent triangles)
In triangles $A O B$ and $C O D$,
$\mathrm{OA}$ $=\mathrm{OC}$ (Radii of a circle)
$\mathrm{OB}$ $=\mathrm{OD}$ (Radii of a circle)
$\mathrm{AB}$ $=\mathrm{CD}$ (Given)
Therefore, $\quad \triangle \mathrm{AOB}$ $\cong \Delta \mathrm{COD}$ (SSS rule)
This gives $\quad \angle \mathrm{AOB}$ $=\angle \mathrm{COD}$
(Corresponding parts of congruent triangles)
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