解：$(2) ∠F+∠H$的值不变
由$(1)，$得$∠F=\frac {1}{2} ∠ABC$
∵$∠AGB$与$∠GAB$的平分线交于点$H$
∴$∠AGH= \frac {1}{2} ∠AGB，∠GAH=\frac {1}{2} ∠GAB$
∴$∠H=180°-(∠AGH +∠GAH)=180°- \frac {1}{2} (∠AGB+∠GAB) $
$=180°- \frac {1}{2} (180°-∠ABG)=90°+ \frac {1}{2} ∠ABG$
∴$∠F+∠H= \frac {1}{2} ∠ABC+90°+ \frac {1}{2} ∠ABG=90°+ \frac {1}{2} (∠ABC+∠ABG)=90° +\frac {1}{2} ×180°=180°$
∴$∠F+∠H$的值不变，为$180°$
