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Standard IX

Mathematics

Opposite Sides of a Parallelogram Are Equal

Two circles i...

Question

Two circles intersects at two points $B$ and $C$ . Through $B$ , two line segments $A B D$ and $P B Q$ are drawn to intersects the circles at $A, D$ and $P, Q$ respectively. Prove that $∠ A C P = ∠ Q C D$ .

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Solution

Let these circles intersect at $B$ and $C$
To prove: $∠ A C P = ∠ Q C D$
Proof: $∠ A B P = ∠ Q B D . . . (i)$ (vertically opposite angles)
$∠ A B P = ∠ A C P . . . (i i)$ (angles in the same segment)
Similarly, $∠ Q C D = ∠ Q B D . . . (i i i)$ (angles in the same segment)
From $(i) (i i),$ and $(i i i)$ ,
$∠ A C P = ∠ Q C D$

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Two circles intersects at two points B and C. Through B, two line segments ABD and PBQ are drawn to intersects the circles at A,D and P,Q respectively. Prove that ∠ACP=∠QCD.

Let these circles intersect at B and CTo prove: ∠ACP=∠QCDProof: ∠ABP=∠QBD...(i) (vertically opposite angles)∠ABP=∠ACP...(ii) (angles in the same segment)Similarly, ∠QCD=∠QBD...(iii) (angles in the same segment)From (i)(ii), and (iii),∠ACP=∠QCD

Two circles intersects at two points $B$ and $C$ . Through $B$ , two line segments $A B D$ and $P B Q$ are drawn to intersects the circles at $A, D$ and $P, Q$ respectively. Prove that $∠ A C P = ∠ Q C D$ .