if ∠EBC measures x°, then ∠BED measures x°.
∠HEF has the same measure as ∠EBC.
The sum of the measures of ∠GHE and ∠DEH is 180°.
The alternate interior angles theorem states that when two parallel lines are cut through by a transversal, the alternate interior angles that are formed are congruent.
By applying the alternate interior angles theorem to parallel lines DF and AC cut through by transversal BE, we have the following pair of congruent angles;
m∠EBC ≅ m∠BED = x°
By applying corresponding angles theorem to the two parallel lines GI and DF, we have the following congruent angles:
m∠HEF ≅ m∠EBC = x°.
By applying the converse of same side interior angles postulate, we have the following supplementary angles:
m∠GHE + m∠DEH = 180°.