To solve the problem, let’s analyze the given setup step by step:

Information from the Problem:

1. Shape: A rectangle $EFGH$.
2. Given angle: $m\angle GIH = 54^\circ$.
3. Objective: Find $m\angle EIF$.

Key Observations:

1. In the rectangle $EFGH$, diagonals $EG$ and $FH$ intersect at point $I$.
2. Since diagonals of a rectangle bisect each other, point $I$ is the midpoint of both diagonals.
3. Diagonals of a rectangle divide the rectangle into four right triangles.
4. Triangle $\triangle GIH$ is one such right triangle.

Relationship Between Angles:

• In $\triangle GIH$:
  • $m\angle GIH = 54^\circ$ is given.
  • Since it is a right triangle ($\angle GIH + \angle IGH + \angle HGI = 90^\circ$): $m\angle IGH = 90^\circ - 54^\circ = 36^\circ$
• Diagonal $EG$ extends across the rectangle:
  • $m\angle EIF$ is directly opposite to $m\angle IGH$, as they are vertically opposite angles.

Conclusion:

$m\angle EIF = m\angle IGH = 36^\circ$

Would you like a detailed geometric proof or clarification on any step? Here are some related questions for exploration:

1. Why do diagonals of a rectangle bisect each other?
2. How does the property of right triangles help determine the angle?
3. What is the sum of all angles in a rectangle?
4. Why are $\angle EIF$ and $\angle IGH$ vertically opposite?
5. How would the solution change if $EFGH$ was not a rectangle?

Tip: Always double-check triangle relationships, especially in problems involving diagonals.