Mathematics
In the figure (ii) given below, ABCD is a cyclic quadrilateral. The tangent to the circle at B meets DC produced at F. If ∠EAB = 85° and ∠BFC = 50°, find ∠CAB.
Circles
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Answer
ABCD is a cyclic quadrilateral.
In cyclic quadrilateral, the exterior angle = opposite interior angle.
∴ ∠BCD = ∠EAB = 85°
From figure,
⇒ ∠BCD + ∠BCF = 180° (∵ both are linear pair)
⇒ ∠BCF + 85° = 180°
⇒ ∠BCF = 95°.
Now in △BCF,
Since, sum of angles in a triangle is 180°.
⇒ ∠BCF + ∠BFC + ∠CBF = 180°
⇒ 95° + 50° + ∠CBF = 180°
⇒ ∠CBF + 145° = 180°
⇒ ∠CBF = 35°.
We know, BF is a tangent and BC is a chord.
∴ ∠CAB = ∠CBF = 35° (∵ angles in alternate segment are equal.)
⇒ ∠CAB = 35°.
Hence, the value of ∠CAB = 35°.
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