In the figure $∠ A B C = 135^{\circ}, ∠ A B X = 90^{\circ}, ∠ X C D = 55^{\circ}, ∠ B C D = 100^{\circ}$ ,then determine whether $∠ X B C$ and $∠ X C B$ are congruent to each other.

∠ X B C = 45^{\circ}, ∠ X C B = 46 \circ

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∠ X B C = 45^{\circ}, ∠ X C B = 45^{\circ}

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∠ X B C = 45^{\circ}, ∠ X C B = 60^{\circ}

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∠ X B C = 45^{\circ}, ∠ X C B = 25^{\circ}

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Solution

The correct option is C $∠ X B C = 45^{\circ}, ∠ X C B = 45^{\circ}$
Given,
$∠ A B C = 135^{o}, A B X = 90^{o}, X C D = 55^{o}, B C D = 100^{o}$ .
Here, $∠ A B C = ∠ A B X + ∠ X B C \Rightarrow X B C = ∠ A B C - A B X = 135^{o} - 90^{o} \Rightarrow X B C = 45^{o} .$
Again $∠ B C D = ∠ X C B + ∠ X C D \Rightarrow ∠ X C B = ∠ B C D - ∠ X C D \Rightarrow ∠ X C B = 100^{o} - 55^{o}$ $= 45^{o}$ .

Since, $∠ X B C = ∠ X C B = 45^{o}$ , they are congruent.

Hence, option $B$ is correct.

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