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In the figure (i) given below, AT is tangent to a circle at A. If ∠BAT = 45° and ∠BAC = 65°, find ∠ABC.

In the figure (i) given below, AT is tangent to a circle at A. If ∠BAT = 45° and ∠BAC = 65°, find ∠ABC. Circles, ML Aggarwal Understanding Mathematics Solutions ICSE Class 10.

Circles

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Answer

From figure,

∠ACB = ∠BAT = 45° (∵ angles in alternate segments are equal)

Since sum of angles in a triangle = 180°.

In △ABC,

⇒ ∠ACB + ∠CAB + ∠ABC = 180°
⇒ 45° + 65° + ∠ABC = 180°
⇒ 110° + ∠ABC = 180°
⇒ ∠ABC = 180° - 110°
⇒ ∠ABC = 70°.

Hence, the value of ∠ABC = 70°.

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