Prove that ∠ACP = ∠QCD."

Academic Mathematics NCERT Class 9

Given :

Two circles intersect at B and C.

Line segments ABD and PBQ are intersecting the circles at A, D , P , Q respectively.

To Prove :

$$\displaystyle \angle ACP\ =\ \angle QCD$$

Construction :

Join AP and QD

$\displaystyle \begin{array}{{>{\displaystyle}l}}
\angle ACP\ =\ 1\ and\ \ \angle QCD\ =\ 2\\
\\
\angle ABP\ =\ 3\ and\ \ \angle QBD\ =\ 4
\end{array}$

Proof :

For chord AP ,

$\displaystyle \begin{array}{{>{\displaystyle}l}}
\angle 1\ and\ \angle 3\ \ lie\ on\ the\ same\ segment\ ACBPA\\
\\
so,\ \angle 1\ =\ \angle 3.............................( i)
\end{array}$

(Angles in the same segment are always equal )

For chord DQ ,

$\displaystyle \begin{array}{{>{\displaystyle}l}}
\angle 2\ and\ \angle 4\ \ lie\ on\ the\ same\ segment\ DQCBD\\
\\
so,\ \angle 2\ =\ \angle 4.............................( ii)
\end{array}$

Lines ABD and PBQ intersect at B

$\displaystyle so,\ \angle 3\ =\ \angle 4.............................( iii)$

(Vertically opposite angles are equal)

From (i) , (ii) and (iii)

$\displaystyle \begin{array}{{>{\displaystyle}l}}
\ \angle 1\ =\ \angle 2\\
\\
\angle ACP\ =\ \angle QCD\ \ \ \ \
\end{array}$

Hence proved.