Question

ΔABC is an isosceles triangle with AB = AC. GB and HC ARE perpendiculars drawn on BC.

Prove that 
(i) BG = CH
(ii) AG = AH

Answer 1

In ΔABC
AB = AC
∠ABC = ∠ACB  ...(equal sides have equal angles opposite to them)...(i)
∠GBC = ∠HCB = 90°   ........(ii)
Subtracting (i) from (ii)
∠GBA = ∠HCA..........(iii)
In ΔGBA and ΔHCA
∠GBA = ∠HCA  ...(from iii)
∠BAG - ∠CAH   ...(vertically opposite angles)
BC = BC
Therefore, ΔGBA ≅ ΔHCA  ...(ASA criteria)
Hence, BG = CH and AG = AH.