In the given figure, If $A B \| C D, E F \perp C D$ and $\angle G E D=126^{\circ}$, find $\angle A G E, \angle G E F$ and $\angle F G E$.

Solution:

It is given that,

$A B \| C D$

$\mathrm{EF} \perp \mathrm{CD}$

$\angle G E D=126^{\circ}$

$\Rightarrow \angle G E F+\angle F E D=126^{\circ}$

$\Rightarrow \angle G E F+90^{\circ}=126^{\circ}$

$\Rightarrow \angle G E F=36^{\circ}$

$\angle A G E$ and $\angle G E D$ are alternate interior angles.

$\Rightarrow \angle A G E=\angle G E D=126^{\circ}$

However, $\angle A G E+\angle F G E=180^{\circ}$ (Linear pair)

$\Rightarrow 126^{\circ}+\angle \mathrm{FGE}=180^{\circ}$

$\Rightarrow \angle \mathrm{FGE}=180^{\circ}-126^{\circ}=54^{\circ}$

$\therefore \angle \mathrm{AGE}=126^{\circ}, \angle \mathrm{GEF}=36^{\circ}, \angle \mathrm{FGE}=54^{\circ}$