In the figure, ∠BAD=70∘, ∠ABD=56∘and ∠ADC=72∘. Which of the following is the measure of ∠BDC, ∠BCD and ∠CBD?

∠ BDC = 18 ∘ , ∠ BCD = 110 ∘ , ∠ CBD = 52 ∘

In the figure, ∠BAD=70∘, ∠ABD=56∘and ∠ADC=72∘. Which of the following is the measure of ∠BDC, ∠BCD and ∠CBD?

∠ BDC = 18 ∘ , ∠ BCD = 110 ∘ , ∠ CBD = 52 ∘

∠ BDC = 18 ∘ , ∠ BCD = 110 ∘ , ∠ CBD = 54 ∘

∠ BDC = 18 ∘ , ∠ BCD = 120 ∘ , ∠ CBD = 52 ∘

∠ BDC = 10 ∘ , ∠ BCD = 110 ∘ , ∠ CBD = 52 ∘

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In the figure, $∠ BAD = 70^{\circ}$ , $∠ ABD = 56^{\circ}$ and $∠ ADC = 72^{\circ}$ . Which of the following is the measure of $∠ BDC$ , $∠ BCD$ and $∠ CBD$ ?

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∠ BDC = 18^{\circ}

∠ BCD = 110^{\circ}

∠ CBD = 52^{\circ}

∠ BDC = 18^{\circ}

∠ BCD = 110^{\circ}

∠ CBD = 54^{\circ}

∠ BDC = 18^{\circ}

∠ BCD = 120^{\circ}

∠ CBD = 52^{\circ}

∠ BDC = 10^{\circ}

∠ BCD = 110^{\circ}

∠ CBD = 52^{\circ}

answer is A.

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Detailed Solution

Consider the given question,
Question Image

Given,

∠ BAD = 70^{\circ}

∠ A B D = 56^{\circ}

and

∠ ADC = 72^{\circ}

.
ABCD is a cyclic quadrilateral.
We know that,
Sum of opposite angles of a cyclic quadrilateral is

180^{o}

.
Therefore,

∠ BAD + ∠ BCD = 180^{\circ}

∠ BCD = 180^{\circ} - ∠ BAD

∠ BCD = 180^{\circ} - 70^{\circ}

∠ BCD = 110^{\circ}

.....(i).
Also,

∠ ADC + ∠ ABC = 180^{o}

72^{\circ} + (56^{\circ} + ∠ CBD) = 180^{\circ}

∠ CBD = 180^{\circ} - 128^{\circ}

∠ CBD = 52^{\circ}

.....(ii).
In

△ BCD

∠ BCD + ∠ DBC + ∠ BDC = 180^{\circ}

...[Angle sum property].

110^{\circ} + 52^{\circ} + ∠ BDC = 180^{\circ}

∠ BDC = 18^{\circ}

...(iii).
Hence, the measure of

∠ BDC

∠ BCD

and

∠ CBD

are

18^{\circ}

110^{\circ}

and

52^{\circ}

.
Therefore, option 1 is correct.

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