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Theorem on Cyclic Quadrilateral and Its Converse

In the given ...

Question

In the given figure,C and D are points on a semicircle described on AB as diameter. $∠ A B D = 75^{\circ}$ and $∠ D A C = 35^{\circ}$ . What is $∠ B D C ?$

130^{\circ}

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110^{\circ}

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90^{\circ}

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100^{\circ}

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Solution

The correct option is A $130^{\circ}$
$∠ A D B = 90$

$I n △ A D B$

$∠ D A B = 180 - 90 - 75 = 15$ (Angle on the Diameter)

$∠ C A B = ∠ C A D + ∠ D A B = 35 + 15 = 50$

$A B C D$ is a cyclic quadrilateral which implies opposite angles are supplementary

$∠ C D B + ∠ C A B = 180$

$∠ C B D = 180 - 50 = 130^{\circ}$

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Similar questions

Q. In the figure,

C

and

D

are the points on the semicircle inscribed on

B A

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∠

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= 70^{o}

and

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In the given figure, C and D are points on the semi-circle described on AB as diameter. Given angle BAD = $70^{0}$ and angle DBC= $30^{0}$ . Calculate $∠ B D C$

In the given figure, if $A E ∥ D C$ and $A B = A C,$ the value of $∠ A B D$ is

(a) $70^{\circ}$

(b) $110^{\circ}$

(d) $130^{\circ}$

Q. In the given figure,

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Q. In the given figure, AOB is a diameter of the circle and C, D, E are any three points on the semicircle. Find the value of ∠ACD + ∠BED.

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In the given figure,C and D are points on a semicircle described on AB as diameter. ∠ABD=75∘ and ∠DAC=35∘. What is ∠BDC?

The correct option is A 130∘∠ADB=90 In△ADB ∠DAB=180−90−75=15 (Angle on the Diameter) ∠CAB=∠CAD+∠DAB=35+15=50 ABCD is a cyclic quadrilateral which implies opposite angles are supplementary ∠CDB+∠CAB=180 ∠CBD=180−50=130∘

In the given figure,C and D are points on a semicircle described on AB as diameter. $∠ A B D = 75^{\circ}$ and $∠ D A C = 35^{\circ}$ . What is $∠ B D C ?$