Angle <EFGH is a trapezoid
angle <H=65 degrees
angle <HEG=77 degrees
Calculate angle <FGE
Since <EFGH is a trapezoid, we know that opposite angles are supplementary. Therefore,
<FGH = 180 - <EFG
Also, angle <GEH is supplementary to <HEG, so
<GEH = 180 - <HEG
Adding these two equations gives us:
<FGH + <GEH = 180 - <EFG + 180 - <HEG
Simplifying:
<FGH + <GEH = 360 - (<EFG + <HEG)
<FGH + <GEH = 360 - (77 + <EFG)
Substituting in the given value <H = 65:
65 + 77 + <FGE + <EGF = 360 - (77 + <EFG)
142 + <FGE + <EGF = 360 - <EFG - 77
219 + <FGE + <EGF = 360 - <EFG
72 + <FGE + <EGF = -<EFG
<EGF = -72 - <FGE + <EFG
Since the sum of the angles of a quadrilateral is 360 degrees, we know that:
<FGE + <EGF + <GEH + <HEG + <FGH + <H = 360
Substituting in the given values and simplifying:
<FGE + (-72 - <FGE + <EFG) + (180 - 77) + 77 + <FGH + 65 = 360
123 + <FGE - <EFG + <FGH = 360
<FGH = 212 - <FGE + <EFG
Now that we have two equations for <FGH (from earlier) and <FGH (just found), we can set them equal to each other and solve for <FGE:
180 - <EFG = 212 - <FGE + <EFG
<FGE = 32 degrees
Therefore, angle <FGE is 32 degrees.