Given that \(HI \perp IJ\) and \(FG \perp EF\), and the angles \(HIJ\) and \(EFG\) are both right angles, it follows from the transitive property that \(HIJ\) is congruent to \(EFG\) by the definition of congruent angles.
The provided statements are:
1. \(HI \perp IJ\) (Given)
2. \(FG \perp EF\) (Given)
3. \(m\angle HIJ = 90\) (Definition of perpendicular lines)
4. \(m\angle EFG = 90\) (Definition of perpendicular lines)
5. \(m\angle HIJ = m\angle EFG\) (Transitive property of equality)
6. \(m\angle HIJ \cong m\angle EFG\) (Definition of congruent angles)
So, the missing reason (f) should be "Definition of congruent angles" to complete the proof. The proof is now logically consistent and complete.