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Model Paper 6 — Maths STD 9 — Question

Gujarat BoardEnglish MediumSTD 9MathsModel Paper 62 Marks
Question
Two circles intersect at two points $B$ and $C.$ Through $B,$ two line segments $\ce{ABD}$ and $\ce{PBQ}$ are drawn to intersect the circles at $\ce{A, D, P, Q}$ respectively $($see figure$).$ Prove that $\ce{ACP=QCD}$.
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Answer

In triangles $A C D$ and $Q C P$,
$\angle A =\angle P$ and $\angle Q =\angle D [$Angles in same segment$]$
$\therefore \angle ACD=\angle QCP [$Third angles$] ...(i)$
Subtracting $\angle PCD$ from both the sides of $eq. (i),$ we get,
$\angle ACD =\angle PCD =\angle QCP -\angle PCD$
$\angle ACP =\angle QCD $
Hence proved.

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