MathematicsIn the figure AD||BC; ∠CAB=45∘; ∠DBC=55∘, then which of the following is the measure of ∠DCB?

In the figure $AD | | BC; ∠ CAB = 45^{\circ}; ∠ DBC = 55^{\circ}$ , then which of the following is the measure of $∠ DCB$ ?

Consider the given figure,

From the figure,

AD | | BC; ∠ CAB = 45^{\circ}; ∠ DBC = 55^{\circ}

.
Here,

∠ DAC = ∠ DBC = 55^{o}

...(angles in the same segment).
Then,

∠ DAB = ∠ DAC + ∠ CAB

∠ DAB = 55^{o} + 45^{o}

∠ DAB = 100^{o}

Now,In a cyclic quadrilateral sum of opposite angles are supplementary.
Then,

∠ DAB + ∠ DCB = 180^{o}

∠ DCB = 180^{o} - ∠ DAB

\Rightarrow ∠ DCB = 180^{o} - 100^{o}

\Rightarrow ∠ DCB = 80^{o}

Hence, the measure of

∠ DCB

80^{o} .

Therefore, option 2 is correct.

JEE