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In the given figure, $ABCD$ is a cyclic quadrilateral in which $∠ CAD = 25^{o}, ∠ ABC = 50^{o}$ and $∠ ACB = 35^{o}$ . Then, which of the following are the measure of $∠ CBD$ , $∠ DAB$ and $∠ ADB$ respectively?

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∠ CBD = 35^{o}

∠ DAB = 80^{o}

∠ ADB = 85^{o}

∠ CBD = 25^{o}

∠ DAB = 35^{o}

∠ ADB = 85^{o}

∠ CBD = 25^{o}

∠ DAB = 70^{o}

∠ ADB = 35^{o}

∠ CBD = 85^{o}

∠ DAB = 70^{o}

∠ ADB = 95^{o}

answer is C.

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Detailed Solution

Given that,

ABCD

is a cyclic quadrilateral such that

∠ CAD = 25^{o}, ∠ ACB = 35^{o}, ∠ ABC = 5 0^{o} .

Here,

∠ CBD

∠ CAD

are angles subtended by the chord

CD

to the circumference.

∴ ∠ CBD = ∠ CAD = 25^{o}

... [since the angles subtended by a chord of a circle to the circumference of the same circle are equal].
Similarly,

∠ ACB

∠ ADB

are angles subtended by the chord

AB

to the circumference.

∴ ∠ ACB = ∠ ADB = 35^{o}

Also,

∠ ADC

∠ ABC

are angles subtended by the chord

AB

to the circumference.

∴ ∠ ADC = ∠ ABC = 50^{o}

From the figure,

∠ BDC = ∠ ADC + ∠ ADB

∠ BDC = 50^{o} + 35^{o}

∠ BDC = 85^{o}

Now,

ABCD

is a cyclic quadrilateral.

∴

The sum of its opposite angles

= 180^{o} .

∠ BAC + ∠ BDC = 180^{o}

⟹ ∠ BAC = 180^{o} - ∠ BDC

⟹ ∠ BAC = 180^{o} - 85^{o}

⟹ ∠ BAC = 95^{o} .

∴ ∠ DAB = ∠ BAC - ∠ CAD

∠ DAB = 95^{o} - 25^{o}

∠ DAB = 70^{o} .

Hence, the measure of angle

∠ CBD = 25^{o}

∠ DAB = 70^{o}

∠ ADB = 35^{o}

.
Therefore, option 3 is correct.

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In the given figure, ABCD is a cyclic quadrilateral in which ∠CAD=25o,∠ABC=50oand ∠ACB=35o. Then, which of the following are the measure of ∠CBD, ∠DAB and ∠ADB respectively?

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In the given figure, $ABCD$ is a cyclic quadrilateral in which $∠ CAD = 25^{o}, ∠ ABC = 50^{o}$ and $∠ ACB = 35^{o}$ . Then, which of the following are the measure of $∠ CBD$ , $∠ DAB$ and $∠ ADB$ respectively?