A triangle ABC is inscribed in a circle, and the tangent at C to the circle is parallel to the bisector of angle ABC. Find the magnitude of $∠ A B C$

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Solution

According to the alternate segment theorem, angle between a tangent and a chord through a point of contact is equal to the angle in the alternate segment.

$⟹ ∠ B C X = ∠ B A C = 40^{\circ}$

Given $B D ∥ X Y$

$⟹ ∠ D B C = ∠ B C X = 40^{\circ}$

Given that BD is angle bisector of angle ABC

$⟹ ∠ A B D = ∠ D B C = 40^{\circ}$

$⟹ ∠ A B C = 2 ∠ D B C = 80^{\circ}$

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A triangle ABC is inscribed in a circle, and the tangent at C to the circle is parallel to the bisector of angle ABC. Find the magnitude of ∠ABC

A triangle ABC is inscribed in a circle, and the tangent at C to the circle is parallel to the bisector of angle ABC. Find the magnitude of $∠ A B C$