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In the given figure, we have $∠ A B C = ∠ A C B$ and $∠ 3 = ∠ 4$ . Find the relation between BD and DC.

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BD = DC

BD > DC

BD < DC

BD = AB

answer is A.

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Detailed Solution

Given a figure, in which

∠ A B C = ∠ A C B

and

∠ 3 = ∠ 4

From the figure,
The point B consists of two angles ∠1 and ∠4. Add the angle,

∠ ABC = ∠ 1 + ∠ 4

∠ 1 = ∠ ABC - ∠ 4

...... (1)
The point C consists of two angles ∠2 and ∠3. Add the angle,

∠ ACB = ∠ 3 + ∠ 2

∠ 2 = ∠ ACB - ∠ 3

….. (2)
By using Euclid’s axiom theorem, if equals be subtracted from equals, the remainders are equal.
From (1) and (2),

∠ ABC - ∠ 4 = ∠ ACB - ∠ 3

Since,

∠ ABC = ∠ ACB

and

∠ 3 = ∠ 4

. So, ∠1 = ∠2.
Therefore, BD = DC.
Hence, option (1) is correct.

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