For the given figure, we will prove the triangles (BGH) and (BDH) are congruent
As shown in the figure:
ΔFGE is an equilateral triangle because the measure of its angles are equal
So, FG = GE
The triangles (GEH) and (DEH) have the following:
1) GE = DE ( because FG = GE and FG = DE )
2) m∠GEH = m∠DEH ( given as shown )
3) EH = EH ( reflexive property )
So, ΔGEH ≅ Δ DEH by SAS
As a result of CBCTC:
GH = DH
m∠EHG = m∠DHE
And The angle EHB has a measure of 180 (straight line angle )
So, m∠GHB = m∠DHB
So, the triangles (BGH) and (BDH) have the following:
1) GH = DH (proved from the congruent triangles (GEH) and (DEH) )
2) m∠GHB = m∠DHB ( proved )
3) BH = BH ( reflexive property )
So,
ΔBGH ≅ Δ BDH by SAS