In the given figure, ABCD is a cyclic quadrilateral whose diagonals intersect each other at O. If ∠ACB=500and ∠ABC=1100, then which of the following is the measure of ∠BDC?

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A
$110^{0}$
B
$60^{0}$
C
$40^{0}$
D
$20^{0}$

Solution:

Consider the given figure,

From the figure-

ABCD

is a cyclic quadrilateral whose diagonals intersect each other at

O

. Also,

∠ ACB = 50^{0}

and

∠ ABC = 110^{0} .

Since

∠ ADB

and

∠ ACB

are formed on same base

AB

, so

∠ ADB = ∠ ACB

⟹

∠ ADB = 50^{0}

...[because

∠ ACB = 50^{0}

]. Sum of opposite angles of cyclic quadrilateral is

180^{0}

. Then,

∠ ADC + ∠ ABC = 180^{0}

⟹

∠ ADC + 110^{0} = 180^{0}

....[because

∠ ABC = 110^{0}]

⟹

∠ ADC = 180^{0} - 110^{0} = 70^{0}

Now,

∠ ADC = ∠ ADB + ∠ BDC

⟹

70^{0} = 50^{0} + ∠ BDC

⟹

70^{0} - 50^{0} = ∠ BDC

⟹

20^{0} = ∠ BDC

Hence, the measure of

∠ BDC

20^{0}

.
Therefore, option 4 is correct.

In the given figure, $ABCD$ is a cyclic quadrilateral whose diagonals intersect each other at $O$ . If $∠ ACB = 50^{0}$ and $∠ ABC = 110^{0}$ , then which of the following is the measure of $∠ BDC$ ?

Solution:

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