An exterior angle of a triangle is $110^{\circ},$ and one of its interior opposite angles is $30^{\circ} .$ Find the other two angles of the triangle.

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$80^{\circ}$ and $70^{\circ}$

$90^{\circ}$ and $80^{\circ}$

$70^{\circ}$ and $60^{\circ}$

$60^{\circ}$ and $50^{\circ}$

answer is B.

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Detailed Solution

Let $∠ B = 30^{\circ}$

$∠ A C D = ∠ B + ∠ A$ (By exterior angle property of a triangle)
$\Rightarrow 110^{\circ} = 30^{\circ} + ∠ A$
$\Rightarrow ∠ A = 110^{\circ} - 30^{\circ}$

$\Rightarrow ∠ A = 80^{\circ}$

In $Δ A B C,$ we have
$∠ A + ∠ B + ∠ C = 180^{\circ}$ (By angle sum property of a triangle)
$\Rightarrow 80^{\circ} + 30^{\circ} + ∠ C = 180^{\circ}$
$\Rightarrow ∠ C = 180^{\circ} - 80^{\circ} - 30^{\circ}$

$\Rightarrow ∠ C = 180^{\circ} - 110^{\circ}$
$\Rightarrow ∠ C = 70^{\circ}$
Therefore, the other two angles of the triangle are $80^{\circ}$ and $70^{\circ}$ .