If ABCD is a cyclic quadrilateral with AB as the diameter of the circle and $∠ A C D = 50^{\circ},$ then $∠ B A D$ = ___.

$70^{\circ}$

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$80^{\circ}$

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$50^{\circ}$

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$40^{\circ}$

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Solution

The correct option is C $40^{\circ}$

Since an angle in a semicircle is a right angle, $∠ A C B = 90^{\circ} .$
But, $∠ B C D = ∠ A C D + ∠ A C B = 50^{\circ} + 90^{\circ} = 140^{\circ} .$
Consider the cyclic quadrilateral ABCD. Since opposite angles are supplementary,
$∠ B C D + ∠ D A B = 180^{\circ} .$
$⟹ ∠ B A D = 180^{\circ} - 140^{\circ} = 40^{\circ}$

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If ABCD is a cyclic quadrilateral with AB as the diameter of the circle and ∠ACD=50∘, then ∠BAD = ___.

The correct option is C 40∘Since an angle in a semicircle is a right angle, ∠ACB=90∘. But, ∠BCD=∠ACD+∠ACB=50∘+90∘=140∘. Consider the cyclic quadrilateral ABCD. Since opposite angles are supplementary, ∠BCD+∠DAB=180∘. ⟹∠BAD=180∘−140∘=40∘

If ABCD is a cyclic quadrilateral with AB as the diameter of the circle and $∠ A C D = 50^{\circ},$ then $∠ B A D$ = ___.