**Problem:**
In the figure, AB ∥ CD, EF ⟂ CD and ∠GED = 126°. Find ∠AGE, ∠GEF and ∠FGE.
**Solution:**
Since AB ∥ CD and GE is a transversal:
$$\angle AGE = \angle GED = 126^\circ$$
Because EF ⟂ CD:
$$\angle FED = 90^\circ$$
Also:
$$\angle GED = \angle GEF + \angle FED$$
Substituting values:
$$126^\circ = \angle GEF + 90^\circ$$
Therefore:
$$\angle GEF = 36^\circ$$
Using interior angles on the same side of the transversal:
$$\angle FGE + \angle GED = 180^\circ$$
Thus:
$$\angle FGE + 126^\circ = 180^\circ$$
Hence:
$$\angle FGE = 54^\circ$$
**Final Answers:**
$$\angle AGE = 126^\circ$$
$$\angle GEF = 36^\circ$$
$$\angle FGE = 54^\circ$$
Problem:
In the figure, AB ∥ CD, EF ⟂ CD and ∠GED = 126°. Find ∠AGE, ∠GEF and ∠FGE.
Solution:
Since AB ∥ CD and GE is a transversal:
$$\small \angle AGE = \angle GED = 126^\circ$$
Because EF ⟂ CD:
$$\small \angle FED = 90^\circ$$
Also:
$$\small \angle GED = \angle GEF + \angle FED$$
Substituting values:
$$\small 126^\circ = \angle GEF + 90^\circ$$
Therefore:
$$\small \angle GEF = 36^\circ$$
Using interior angles on the same side of the transversal:
$$\small \angle FGE + \angle GED = 180^\circ$$
Thus:
$$\small \angle FGE + 126^\circ = 180^\circ$$
Hence:
$$\small \angle FGE = 54^\circ$$
Final Answers:
$$\small \angle AGE = 126^\circ$$
$$\small \angle GEF = 36^\circ$$
$$\small \angle FGE = 54^\circ$$
$$
\begin{tikzpicture}[scale=1.2]
% Upper horizontal line AB
\draw[thick] (-2,2) — (3,2);
\node at (-2,2.2) {A};
\node at (0.2,2.2) {G};
\node at (2.8,2.2) {B};
% Lower horizontal line CD
\draw[thick] (-2,0) — (3,0);
\node at (-2,0.2) {C};
\node at (2.8,0.2) {D};
% Vertical EF perpendicular to CD
\draw[thick] (0,2) — (0,0);
\node at (0.2,1.2) {F};
\node at (0.2,-0.3) {E};
% Slanted GE
\draw[thick] (0,0) — (-1,1.5);
% Angle marking for 126 degrees at E
\draw (0,0) + (0:0.6) arc (0:126:0.6);
\node at (0.7,0.5) {$126^\circ$};
\end{tikzpicture}
$$