Question

"Question 9 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 the given figure). Prove that ∠ACP = ∠QCD.

Class 9 - Math - Circles Page 186"

Answer 1

Hi friend ✋

Solution---Since, angles in the same segment are equal.

angle ACP=angleABP......(3)

QCD=QBD.........(2)

angleABP=angleQBD(V.o.a)...….(3)

From 1,2 ,3 we have..

angleACP =angle QCD

Hope it will helpful.....

Answer 2

Solution:

Given: Two circles intersect at two points B & C. Through B, two line segments ABD & PBQ are drawn which intersect the Circles at A, D, P & Q.

To Prove:
∠ACP = ∠QCD

Proof:

In circle I,
For chord AP,
∠PBA = ∠ACP (Angles in the same segment are equal) — (i)

In circle II,

For chord DQ,
∠DBQ = ∠QCD (Angles in same segment) — (ii)

ABD and PBQ are line segments intersecting at B.

∠PBA = ∠DBQ (Vertically opposite angles) —iii
From the equations (i), (ii) and (iii),

∠ACP = ∠QCD

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Hope this will help you.....