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We are tasked with writing a paragraph proof to show that \( \angle DHF \cong \angle GHF \) given the information provided:

### Given:
- \( m\angle ABC = 114^\circ \)
- \( m\angle DHE = 25^\circ \)
- \( m\angle EHF = 41^\circ \)
- \( \angle ABC \) and \( \angle GHF \) are supplementary

### To Prove:
- \( \angle DHF \cong \angle GHF \)

### Paragraph Proof:
We begin by considering the fact that \( \angle ABC \) and \( \angle GHF \) are supplementary. This means that their measures sum to \( 180^\circ \). Since \( m\angle ABC = 114^\circ \), the measure of \( \angle GHF \) must be:
\[
m\angle GHF = 180^\circ - 114^\circ = 66^\circ.
\]

Next, we examine the angle \( \angle DHF \). From the diagram, \( \angle DHF \) is composed of the angles \( \angle DHE \) and \( \angle EHF \). Therefore, the measure of \( \angle DHF \) is:
\[
m\angle DHF = m\angle DHE + m\angle EHF = 25^\circ + 41^\circ = 66^\circ.
\]

Since \( m\angle DHF = 66^\circ \) and \( m\angle GHF = 66^\circ \), it follows that \( \angle DHF \cong \angle GHF \) by the definition of congruent angles, which states that two angles are congruent if and only if their measures are equal.

Thus, we have proven that \( \angle DHF \cong \angle GHF \).

Would you like more details on any part of this proof, or have any questions?

Here are some related questions:
1. What is the definition of supplementary angles?
2. How do you calculate the measure of an unknown supplementary angle?
3. What is the Angle Addition Postulate?
4. How can you use congruent angles in triangle proofs?
5. What role do angle bisectors play in proving angle congruence?

**Tip**: Always label key information on the diagram when working through a proof—it helps organize thoughts and ensures nothing is missed!

Write a paragraph proof to show that \( \angle DHF \cong \angle GHF \), given the following:
- \( m\angle ABC = 114^\circ \)
- \( m\angle DHE = 25^\circ \)
- \( m\angle EHF = 41^\circ \)
- \( \angle ABC \) and \( \angle GHF \) are supplementary.

Learn how to prove that \( \angle DHF \cong \angle GHF \) using the concept of supplementary angles and angle addition postulate. This problem breaks down the key steps of angle congruence, involving given angle measures of 114°, 25°, and 41°.

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