Given that $m\angle ABC = 114^\circ$, $m\angle DHE = 25^\circ$, and $m\angle EHF = 41^\circ$, and that $\angle ABC$ and $\angle GHF$ are supplementary, we are tasked with proving the measure of $\angle GHF$.

By definition, supplementary angles are two angles whose measures sum up to $180^\circ$. Since it is given that $\angle ABC$ and $\angle GHF$ are supplementary, the sum of their measures must equal $180^\circ$. Thus, we can express this relationship algebraically as:

$m\angle ABC + m\angle GHF = 180^\circ$

Substituting the given measure of $\angle ABC = 114^\circ$ into the equation:

$114^\circ + m\angle GHF = 180^\circ$

To find $m\angle GHF$, subtract $114^\circ$ from both sides:

$m\angle GHF = 180^\circ - 114^\circ = 66^\circ$

Therefore, the measure of $\angle GHF$ is $66^\circ$, satisfying the condition that $\angle ABC$ and $\angle GHF$ are supplementary.

Would you like further details on this proof, or do you have any questions?

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Here are 5 related questions:

1. What is the definition of supplementary angles?
2. How do you find the measure of an unknown angle in a supplementary pair?
3. If two angles are complementary, what is their sum?
4. Can two obtuse angles be supplementary? Why or why not?
5. How does the concept of linear pairs relate to supplementary angles?

Tip: Always verify angle relationships by checking whether their sum matches the property (e.g., 180° for supplementary).