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19/07/2021 1:17 pm
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Two circles intersect at two points B and C. Through B, two line segments ABD and PBQ are drawn to intersect the circles at A, D and P, Q respectively (see Figure). Prove that ∠ACP = ∠QCD.
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19/07/2021 1:18 pm
Construction:
Join the chords AP and DQ.
For chord AP, we know that angles in the same segment are equal.
∠PBA = ∠ACP — (i)
Similarly for chord DQ,
∠DBQ = ∠QCD — (ii)
It is known that ABD and PBQ are two line segments which are intersecting at B.
At B, the vertically opposite angles will be equal.
∴ ∠PBA = ∠DBQ — (iii)
From equation (i), equation (ii) and equation (iii) we get,
∠ACP = ∠QCD