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# In figure, if AB || CD, EF is perpendicular to CD and $$\angle{GED}$$ = $$126^\circ$$ , find $$\angle{AGE}$$,$$\angle{GEF}$$ and $$\angle{FGE}$$

$$\because$$ $$\angle{AGE}$$ = $$\angle{GED}$$ ....(Alternate interior angles)

But it is given that,
$$\because$$ $$\angle{GED}$$ = $$126^\circ$$
$$\therefore$$ $$\angle{AGE}$$ = $$126^\circ$$ ....(i)

Also, $$\angle{GEF}$$ + $$\angle{FED}$$ = $$126^\circ$$
$$\because$$ EF is perpendicular to CD,
$$\therefore$$ $$\angle{GEF}$$ + $$90^\circ$$ = $$126^\circ$$
$$\Rightarrow$$ $$\angle{GEF}$$ = $$36^\circ$$

Also, by linear pair axiom, we get,
$$\angle{AGE}$$ + $$\angle{FGE}$$ = $$180^\circ$$
$$\Rightarrow$$ $$126^\circ$$ + $$\angle{FGE}$$ = $$180^\circ$$ ....(from (i))
$$\therefore$$ $$\angle{FGE}$$ = $$54^\circ$$