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Two Chords Ab and Cd Intersect at P Inside the Circle. Prove that the Sum of the Angles Subtended by the Arcs Ac and Bd at the Centre O is Equal to Twice the Angle Apc. - ICSE Class 10 - Mathematics

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ConceptArc and Chord Properties - If Two Arcs Subtend Equal Angles at the Center, They Are Equal, and Its Converse

Question

Two chords AB and CD intersect at P inside the circle. Prove that the sum of the angles
subtended by the arcs AC and BD at the centre O is equal to twice the angle APC.

Solution

Given : two chords AB and CD intersect each other at P inside the circle. OA,
OB, OC and OD are joined.
To prove: ∠AOC + ∠BOD = 2∠APC
Construction: Join AD.
Proof: Arc AC subtends ∠AOCat the centre and ∠ADCat the remaining
Part of the circle.
∠AOC = 2∠ADC          …………(1)

Similarly,
∠BOD = 2 ∠BAD       ………….(2)
Adding (1) and (2),
∠AOC + ∠BOD =  2∠ADC + 2∠BAD
= 2(∠ADC +∠BAD          ……….(3)
But ΔPAD,
Ext. ∠APC  = ∠PAD +∠ADC
 = ∠BAD +∠ADC          ……………(4)
From (3) and (4),
∠AOC + ∠BOD = 2∠APC

  Is there an error in this question or solution?
Solution Two Chords Ab and Cd Intersect at P Inside the Circle. Prove that the Sum of the Angles Subtended by the Arcs Ac and Bd at the Centre O is Equal to Twice the Angle Apc. Concept: Arc and Chord Properties - If Two Arcs Subtend Equal Angles at the Center, They Are Equal, and Its Converse.
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