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In a triangle ABC, if AB = AC and AB is produced to D such that BD = BC, Then the value of $∠ A C D : ∠ A D C$ is:

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1: 2

2: 4

3: 1

2: 1

answer is C.

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Detailed Solution

Given a triangle ABC in which AB = AC and AB is produced to D such that BD = BC.
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Using angle sum property,

∠ A C B + ∠ B A C + ∠ C B A = 180 °

Also, angles opposite to equal sides are equal
As, AB = AC, therefore

∠ A B C = ∠ A C B … … (1)

In triangle BDC, BD = BC.
Therefore,

∠ B D C = ∠ B C D … … (2)

Using exterior angle property in triangle BDC,

∠ A B C = ∠ B D C + ∠ B C D ⇒ ∠ A B C = ∠ B D C + ∠ B D C ⇒ ∠ A B C = 2 ∠ B D C … … … (3)

Similarly,

∠ A B C = 2 ∠ B C D ⇒ ∠ A C B = 2 ∠ B C D … … . (4)

Now,

∠ A C D = ∠ A C B + ∠ B C D ⇒ ∠ A C D : ∠ A D C = (∠ A C B + ∠ B C D) : ∠ A D C … … (5)

Putting the values,

⇒ ∠ A C D : ∠ A D C = (2 ∠ B D C + ∠ B C D) : ∠ B D C ⇒ ∠ A C D : ∠ A D C = (3 ∠ B D C) : ∠ B D C … … (6)

⇒ ∠ A C D : ∠ A D C = 3 : 1

The ratio is 3: 1.
Hence, option 3 is correct.

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