https://file.aiquickdraw.com/mathbot/file/1728542742685q7d6ptsf.jpg

Let's break down the problem using the given information:

We have the following relationships:

- \( CD \perp EF \) (so \( \angle DCF = 90^\circ \)),
- \( m \angle HCD = 3x - 7 \),
- \( m \angle ECH = x + 5 \),
- We are tasked with finding \( m \angle GCF \).

### Key insights:

1. Since \( CD \perp EF \), we know that the total measure of the angles around point \( C \), which form a straight line, is \( 180^\circ \).
2. \( \angle HCD + \angle ECH + \angle GCF = 90^\circ \), since all these angles are adjacent and add up to the right angle formed by \( CD \perp EF \).

### Step-by-Step Solution:

1. **Sum of angles**:  
   We know:
   \[
   m \angle HCD + m \angle ECH + m \angle GCF = 90^\circ
   \]
   Substituting the expressions for the angles we know:
   \[
   (3x - 7) + (x + 5) + m \angle GCF = 90
   \]
   
2. **Simplifying the equation**:  
   Combine like terms:
   \[
   (3x + x) + (-7 + 5) + m \angle GCF = 90
   \]
   \[
   4x - 2 + m \angle GCF = 90
   \]
   
3. **Solve for \( m \angle GCF \)**:  
   Isolate \( m \angle GCF \) by moving terms involving \( x \) to the other side:
   \[
   m \angle GCF = 90 - (4x - 2)
   \]
   \[
   m \angle GCF = 90 - 4x + 2
   \]
   \[
   m \angle GCF = 92 - 4x
   \]

### Conclusion:
The measure of \( \angle GCF \) is \( 92 - 4x \).

---

Do you want any further details or have any questions?

Here are some related questions you might explore:
1. What happens if we set \( x = 20 \)? What is \( m \angle GCF \)?
2. How would the problem change if \( \angle HCD = 4x - 6 \)?
3. How can we solve for \( x \) if additional information is given?
4. Can you express \( m \angle GCF \) in terms of another variable?
5. What are the properties of perpendicular lines in angle-based problems like this?

**Tip**: When dealing with perpendicular lines, always look for ways to use the fact that they form right angles to simplify calculations.

If CD ⊥ EF, m∠HCD = 3x - 7, and m∠ECH = x + 5, find ∠GCF.

In this geometry problem, we find the measure of ∠GCF, given perpendicular lines CD and EF. Using algebraic expressions for ∠HCD and ∠ECH, the solution walks through solving for ∠GCF in terms of x.

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