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?

see full answer

Talk to JEE/NEET 2025 Toppers - Learn What Actually Works!

Real Strategies. Real People. Real Success Stories - Just 1 call away

An Intiative by Sri Chaitanya

∠ BCD is a right angle.

∠ BCD is an obtuse angle.

∠ BCD is acute angle.

∠ BCD is perpendicular.

answer is A.

(Unlock A.I Detailed Solution for FREE)

Ready to Test Your Skills?

Check your Performance Today with our Free Mock Test used by Toppers!

Take Free Test

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.