Ex.10.5 Q9 Circles Solution - NCERT Maths Class 9

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Question

Two circles intersect at two points \({B}\) and \({C.}\) Through \({B,}\) two line segments \({ABD}\) and \({PBQ}\) are drawn to intersect the circles at \(A, D, P\) and \(Q\) respectively. Prove that \(\begin {align} \angle {ACP}=\angle {QCD.} \end {align}\)

 

 Video Solution
Circles
Ex 10.5 | Question 9

Text Solution

What is given ?

Two circles intersect at two points

What is unknown?

Proof of \( \angle {ACP}=\angle {QCD} \)

Reasoning:

\(\angle {ACP} \)  and  \(\angle {ABP} \) lie on the same segment.

Similarly, \( \angle {DCQ} \)  and  \( \angle {DBQ} \) lie on the same segment.

Angles in the same segment of a circle are equal.

Steps:

We know that, angles in the same segment of a circle are equal.

So we get  \( \angle {ACP}=\angle {ABP} \)  and  \( \angle {QCD}=\angle {QBD} \)

Also, \(\begin {align} \angle {QBD}=\angle {ABP} \end {align}\)(Vertically opposite angles)

Therefore \( \angle {ACP}=\angle {QCD} \)

  
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