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24. In the given figure, C is the centre of the circle. Prove that
anglePCR = 2angleQPR +2angleQRP.
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Given:- In the given fig., O is the centre of circle.
To prove:- ∠AOC=∠AFC+∠AEC
Proof:- In △BEC, Exterior angle at B,
∴∠ABC=∠AEC+∠BCD.....(1)(Exterior angle theorem)
2∠ABC=2∠AEC+∠BCD
∵2∠ABC=∠AOC(Angle subtended on circle is double the angle subtended at centre on same arc)
∴∠AOC=∠AEC+∠BCD+∠AEC+∠BCD
∠AOC=∠AEC+∠BCD+∠ABC(From (1))
∵∠ABC=∠ADC(∵Angle subtends on same arc are equal)
∴∠AOC=∠AEC+∠BCD+∠ADC.....(2)
Now, in △FDC
Exterior angle at F.
∴∠AFC=∠BCD+∠ADC.....(3)
Now, from equation (2)&(3), we have
∠AOC=∠AEC+∠AFC
Hence proved
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