https://file.aiquickdraw.com/mathbot/file/1730308312838s6srdurj.jpg

To find the measure of angle \( \angle GJI \), we need to analyze the given setup.

### Step 1: Identify Parallel Lines and Transversals
The diagram shows two parallel lines \( EF \parallel GH \) (indicated by the arrows on these lines) and a transversal \( CD \) that intersects both parallel lines.

### Step 2: Use the Given Angles
The angles given in the diagram are:
- \( \angle IJG = (12x - 12)^\circ \)
- \( \angle IJC = (4x)^\circ \)

Since \( EF \parallel GH \), the angles \( \angle IJG \) and \( \angle IJC \) are **alternate interior angles** created by the transversal \( CD \). Thus, they are congruent.

### Step 3: Set Up the Equation
Since \( \angle IJG = \angle IJC \), we can write:
\[
12x - 12 = 4x
\]

### Step 4: Solve for \( x \)
1. Subtract \( 4x \) from both sides:
   \[
   8x - 12 = 0
   \]
2. Add 12 to both sides:
   \[
   8x = 12
   \]
3. Divide by 8:
   \[
   x = \frac{12}{8} = 1.5
   \]

### Step 5: Substitute \( x \) to Find \( \angle GJI \)
Now, substitute \( x = 1.5 \) back into either \( \angle IJG \) or \( \angle IJC \) (since they are equal).

Using \( \angle IJG = 12x - 12 \):
\[
\angle IJG = 12(1.5) - 12 = 18 - 12 = 6^\circ
\]

Thus, the measure of \( \angle GJI \) is:
\[
\boxed{6^\circ}
\]

Would you like further details or have any questions?

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Here are five related questions to expand on this problem:
1. What are alternate interior angles, and why are they equal in this scenario?
2. How would the calculation change if \( EF \) and \( GH \) were not parallel?
3. Can you explain other angle relationships formed by a transversal across parallel lines?
4. How can we verify the solution using different methods?
5. What would be the result if one angle was given in terms of another variable?

**Tip:** Always check if lines are marked as parallel, as this often leads to angle relationships such as alternate interior angles being equal.

Solve for the measure of angle ∠GJI in a geometry problem involving parallel lines and a transversal. This problem utilizes algebra and the Alternate Interior Angles Theorem to find the solution step-by-step, making it ideal for students in Grades 7-9.

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