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))
\(\therefore \) \(\angle{FGE}\) = \(54^\circ\)