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prove that the line joining the mid-poin...

prove that the line joining the mid-point of two equal chords of a circle subtends equal angles with the chord.

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Given : A circle with centre O in which two chords `AB=CD`. Also M and N are the mid=points of AB and CD respectively.
To Prove : `angle1 =angel2` and `angle7=angle8`
Construcation: Join OM and ON.
Proof : Since equal chords of a circle are eqidistant from the centre.
`therefore OM=ON`
`rArrangle4-angel3` (angles opposite to equal sides are equal ).....(1)
Also, `angle6=angle5`
(line joining the centre to the mid-point of a chord is perpendicular to the chord)......(2)
Adding (2) and (1),we get
`angle4+angle6=angle3+angle5`
`rArrangle2=angle1`
`rArr180^@-angle2 =180^@-angle1`
`rArrangle8=angle7`
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