UPSC CDS 1 2022 Math Paper

Given:
In a triangle ABC,AB=AC
∠ACD=x
Formula Used:
The sum of the linear pair of angles =180∘
∠A+∠B+∠C=180∘
Calculation:Since, △ABC is an isosceles triangle
So, ∠ABC=∠ACB‌‌[∵AB=AC]‌‌⋯ (i)
According to the figure
∠ACB+∠ACD=180∘
⇒∠ACB+x=180∘
⇒∠ACB=(180∘−x)‌‌−‌ (ii) ‌
From (i) and (ii), we have
⇒∠ABC=(180∘−x)
Now, In △ABC
∠ABC+∠ACB+∠BAC=180∘
⇒(180∘−x)+(180∘−x)+∠BAC=180∘
⇒∠BAC=180∘−360∘+2x
⇒∠BAC=2x−180∘
∴ The value of ∠BAC is 2x−180∘ .