△ABC is an isosceles triangle in which AB=AC. Sides BA is produced to D such that AD=AB. Show that ∠BCD is a right angle.

In△ABC, we haveAB=AC∣given∠ACB=∠ABC... (1)∣Since angles opp. to equal sides are equalNow,AB=AD∣Given∴AD=AC∣SinceAB=ACThus , in△ADC, we haveAD=AC⇒∠ACD=∠ADC... (2)∣Since angles opp. to equal sides are equalAdding (1) and (2) , we get∠ACB+∠ACD=∠ABC+∠ADC⇒∠BCD=∠ABC+∠BDC∣Since∠ADC=∠BDC⇒∠BCD+∠BCD=∠ABC+∠BDC+∠BCD∣Adding∠BCDon both sides⇒2∠BCD=180∘∣Angle sum property⇒∠BCD=90∘Hence,∠BCDis a right angle.