An isosceles triangle ABC has AB = AC. The line AB is produced to a point D, such that DA = AB. Then which of the following options is correct?

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∠ BCD is a right angle.

∠ BCD is an obtuse angle.

∠ BCD is acute angle.

∠ BCD is perpendicular.

answer is A.

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Detailed Solution

Given that, in an isosceles triangle ABC line AB = AC and line DA = AB.
Question Image

Now AB=AC, then
∠ABC=∠ACB……. [1][The angles opposite to equal sides are equal]
In △ACD,
AD = AC
∠ACD=∠ADC……. [2][The angles opposite to equal sides are equal]
In △BCD, the angle sum property of a triangle defines that the sum of all angles of a triangle is 180°.
$∠ B D C + ∠ D B C + ∠ B C D = 1800$
Using equations [1] and [2] we get:
$∠ A C D + ∠ A C B + ∠ B C D = 1800$ ……… [3]
From the figure we have
$∠ A C D + ∠ A C B = ∠ B C D$ ……… [4] On substituting the value of equation [4] in equation [3]
$⇒ ∠ B C D + ∠ B C D = 1800 ⇒ 2 ∠ B C D = 1800 ⇒ ∠ B C D = 900$
Thus, BCD is a triangle with one angle of 90°. Hence, angle BCD is a right angle triangle.
Therefore, option 1 is correct.