To solve for $m \angle EFH$, we can use the information given about the lines and angles in the problem. Here’s the step-by-step solution:

1. Identify Parallel Lines: We know that $\overline{AD} \parallel \overline{EG}$. This implies that angles formed with these lines will be related due to parallel line properties (such as corresponding angles and alternate interior angles).

2. Perpendicular Condition: We are given that $\overline{BH} \perp \overline{FC}$. This tells us that $\angle BHF$ and $\angle FHC$ are both right angles, so $m \angle BHF = 90^\circ$.

3. Angle $m \angle ABH$: It's given that $m \angle ABH = 145^\circ$.

4. Using the Straight Line Property: Since $\angle ABH$ and $\angle EBF$ are on a straight line and $\overline{BH} \perp \overline{FC}$, we can find $m \angle EBF$ by subtracting $m \angle ABH$ from $180^\circ$: $m \angle EBF = 180^\circ - 145^\circ = 35^\circ$

5. Determine $m \angle EFH$: Since $\overline{AD} \parallel \overline{EG}$, $\angle EBF$ and $\angle EFH$ are alternate interior angles. Therefore, $m \angle EFH = m \angle EBF = 35^\circ$.

Thus, the measure of $m \angle EFH$ is: $m \angle EFH = 35^\circ$

Would you like more details or have any questions on the steps?

Here are some related questions to explore:

1. What are corresponding angles and how do they relate to parallel lines?
2. How do perpendicular lines affect the angles they form with other lines?
3. What is the significance of alternate interior angles in parallel lines?
4. How can we apply the straight angle property to solve similar problems?
5. How would the solution change if the given angle were different?

Tip: When dealing with parallel lines, look for alternate interior and corresponding angles to simplify angle calculations.