To solve this problem and prove that $\angle DHF \cong \angle GHF$, let's break down the given information and organize the steps.

Given:

1. $\angle ABC = 114^\circ$
2. $\angle DHE = 25^\circ$
3. $\angle EHF = 41^\circ$
4. $\angle ABC$ and $\angle GHF$ are supplementary, meaning their measures add up to $180^\circ$.

To Prove:

$\angle DHF \cong \angle GHF$

Plan:

We are asked to prove that two angles are congruent, specifically $\angle DHF$ and $\angle GHF$. This involves establishing that their measures are equal.

Steps:

1. Calculate $\angle GHF$: Since $\angle ABC$ and $\angle GHF$ are supplementary, we can calculate the measure of $\angle GHF$ by subtracting the measure of $\angle ABC$ from 180 degrees: $\angle GHF = 180^\circ - 114^\circ = 66^\circ$

2. Calculate $\angle DHF$: $\angle DHF = \angle DHE + \angle EHF$ Substituting the given values: $\angle DHF = 25^\circ + 41^\circ = 66^\circ$

3. Conclusion: Since $\angle DHF = 66^\circ$ and $\angle GHF = 66^\circ$, we conclude that: $\angle DHF \cong \angle GHF$

Two-Column Proof:

Would you like more details or have any questions?

Here are 5 related questions to expand your understanding:

1. How can we identify supplementary angles in different geometric figures?
2. What is the Angle Addition Postulate and how is it used in geometry proofs?
3. What is the significance of congruent angles in triangles?
4. How do we approach solving angle relationships when parallel lines are involved?
5. Can the method of angle addition be applied to three-dimensional shapes?

Tip: In proofs, always clearly label each step and reason to ensure logical flow and clarity.