Question

linear pair
vertical angles theorem
corresponding angles theorem
alternate exterior angles theorem
same - side interior angles
alternate interior angles theorem
$m\angle lfg + m\angle gfk = 180^{\circ}$
$m\angle kfe = m\angle fej$
$m\angle hed = m\angle fej$
$m\angle jef + m\angle lfe = 180^{\circ}$
$m\angle lfg = m\angle deh$
$m\angle dej = m\angle efl$

Explanation:

Step1: Analyze \(m\angle LFG + m\angle GFK=180^{\circ}\)

By the definition of a linear pair (two adjacent angles that form a straight line and their measures add up to \(180^{\circ}\)), this is a linear pair.

Step2: Analyze \(m\angle KFE = m\angle FEJ\)

These are alternate interior angles (angles between two parallel lines and on opposite sides of a transversal), so by the Alternate interior angles theorem, they are equal.

Step3: Analyze \(m\angle HED = m\angle FEJ\)

These are vertical angles (opposite angles formed by the intersection of two lines), so by the Vertical angles theorem, they are equal.

Step4: Analyze \(m\angle JEF + m\angle LFE = 180^{\circ}\)

These are same - side interior angles (angles between two parallel lines and on the same side of a transversal), so by the same - side interior angles theorem, their measures add up to \(180^{\circ}\).

Step5: Analyze \(m\angle LFG = m\angle DEH\)

These are corresponding angles (angles in the same relative position with respect to two parallel lines and a transversal), so by the Corresponding angles theorem, they are equal.

Step6: Analyze \(m\angle DEJ = m\angle EFL\)

These are alternate exterior angles (angles outside two parallel lines and on opposite sides of a transversal), so by the Alternate exterior angles theorem, they are equal.

Answer:

\(m\angle LFG + m\angle GFK = 180^{\circ}\) → linear pair; \(m\angle KFE = m\angle FEJ\) → Alternate interior angles theorem; \(m\angle HED = m\angle FEJ\) → Vertical angles theorem; \(m\angle JEF + m\angle LFE = 180^{\circ}\) → same - side interior angles; \(m\angle LFG = m\angle DEH\) → Corresponding angles theorem; \(m\angle DEJ = m\angle EFL\) → Alternate exterior angles theorem.