Because: Be and CF are bisectors of ∠ABC and ∠ACB, respectively.
So:
∠ABE=∠CBE=( 1/2)∠ABC
∠ACF=∠BCF=( 1/2)∠ACB
1)
So:
∠BGC= 180 -∠CBE-∠BCF
∠BGC = 180-( 1/2)∠ ABC -( 1/2)∠ACB
So:
∠BGC = 180-( 1/2)(∠ABC+∠ACB)
2)
Because: ∠ ABC+∠ ACB = 180-∠ A
Substitute 1) to know:
∠BGC = 180-( 1/2)( 180-∠A)
∠BGC=90 +( 1/2)∠A