ABC is a isosceles triangle in which ab=ac. Side ba produced to d such that ad=an. Show that angle bcd is right angle

Question

Maths class 9 

Answer 1

In △ABC, we have 

AB=AC  ∣ given

∠ACB=∠ABC ... (1) ∣ Since angles opp. to equal sides are equal

Now, AB=AD ∣ Given

∴AD=AC ∣ Since AB=AC

Thus , in △ADC, we have 

AD=AC

⇒∠ACD=∠ADC ... (2) ∣ Since angles opp. to equal sides are equal

Adding (1) and (2) , we get 

∠ACB+∠ACD=∠ABC+∠ADC

⇒∠BCD=∠ABC+∠BDC ∣ Since∠ADC=∠BDC

⇒∠BCD+∠BCD=∠ABC+∠BDC+∠BCD ∣ Adding ∠BCD on both sides

⇒2∠BCD=180∘ ∣ Angle sum property

⇒∠BCD=90∘

Hence, ∠BCD is a right angle.