Question

Using the given, prove that triangle ACF is congruent to triangle ECF by using as many additional steps to prove

Answer 1

△FCA≅△ACF by SAS

BG≅ GD (Definition of isosceles triangle - △BGD)

Since FG bisects ∠HGI, we have ∠GCD=∠GCB=90∘ as GC is an angle bisector

With ∠GCD=∠GCB=90∘ and BC ≅ CD,△GBC is a right-angled isosceles triangle.
Since △GBC is a right-angled isosceles triangle and FG bisects ∠HGI, ∠FCA is congruent to ∠ACF by SAS congruence (Side-Angle-Side, using the right-angled isosceles triangle and the bisected angle).

Answer 2

Clearly, the angle equality implies that GBD is isosceles so GB=GD. Thus BC=CD since GC is an angle bisector. And GCD=GCB=90 as well since GC is an angle bisector. Thus, FCA is congruent to ACF by SAS congruency.