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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.

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, ML Aggarwal Understanding Mathematics Solutions ICSE Class 10.

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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