To solve for $m\angle F$ in the triangle $\triangle EFG$, let’s analyze the information provided:

1. Given:
   • $\angle H = 58^\circ$
   • The markings on $EH$ and $HF$ (indicating congruent segments) suggest isosceles triangle properties in $\triangle EHF$.

Step-by-step solution:

1. Identify isosceles triangle property: Since $EH \cong HF$, $\triangle EHF$ is isosceles. In an isosceles triangle, the base angles are congruent.

   Therefore: $\angle E = \angle F$

2. Sum of angles in a triangle: The sum of all angles in a triangle is $180^\circ$. For $\triangle EHF$: $\angle E + \angle F + \angle H = 180^\circ$

3. Substitute known values: We know $\angle H = 58^\circ$ and $\angle E = \angle F$. Let $\angle F = x$. Then: $x + x + 58^\circ = 180^\circ$

   Combine like terms: $2x + 58^\circ = 180^\circ$

4. Solve for $x$: Subtract $58^\circ$ from both sides: $2x = 122^\circ$

   Divide by 2: $x = 61^\circ$

Final Answer:

The measure of $\angle F$ is: $\boxed{61^\circ}$

Let me know if you'd like me to clarify further or go over any steps in detail!

Related Questions:

1. How do you identify an isosceles triangle based on its side markings?
2. What is the sum of angles in any triangle?
3. How can you determine congruent angles in an isosceles triangle?
4. What is the process to find missing angles in a triangle using algebra?
5. How would the solution change if $\angle H$ had a different value?

Tip:

Always look for congruent sides or angles when working with triangles—these clues often simplify the problem significantly!