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Question
In a triangle ABC, if AB = AC and AB is produced to D such that BD = BC, find ∠ACD: ∠ADC.
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Solution
In the given ΔABC,AB = ACand AB is produced to D such that BD = BC
We need to find ∠ACD : ∠ADC
Now, using the property, “angles opposite to equal sides are equal”
As AB = AC
∠6 = ∠4 ........(1)
Similarly,
As BD = BC
∠1 = ∠2 ........(2)
Also, using the property, “an exterior angle of the triangle is equal to the sum of the two opposite interior angle”
In ΔBDC
ext. ∠6 = ∠1 + ∠2
ext. ∠6 = ∠1 + ∠1 (Using 2)
ext. ∠6 = 2∠1
From (1), we get
ext. ∠4 = ∠2 .......(3)
Now, we need to find ∠ACD : ∠ADC
That is,
(∠4 + ∠2): ∠1
(2∠1 + ∠2) : ∠1 (Using 3)
(2∠1 + ∠1) : ∠1(Using 2)
3∠1 :∠1
Eliminating ∠1from both the sides, we get 3:1
Thus, the ratio of ∠ACD :∠ADC is 3 :1
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