https://file.aiquickdraw.com/mathbot/file/1729532480960de9fyf86.jpg

Let's analyze the given image.

The figure shows a quadrilateral \( DEFG \), where diagonals \( DF \) and \( EG \) intersect at point \( P \). The side lengths are labeled as follows:
- \( DE = 4x \) inches
- \( EF = 4 \) inches
- \( FG = 3x + 7 \) inches
- \( GD = x + 9 \) inches

The question asks to find the relationship between \( DE \) and the other sides of the quadrilateral. We are tasked with completing the equation \( DE \cong \) [choose an option] and solve for \( DE \).

### Step 1: Identify the relationship
In this case, the question is suggesting that two sides of the quadrilateral are congruent. This is most likely a parallelogram, where opposite sides are congruent. If \( DEFG \) is a parallelogram, the opposite sides are congruent:
- \( DE \cong GF \)
- \( EF \cong DG \)

### Step 2: Solve for \( x \)
Since \( DE \cong GF \), we can set up the equation:
\[
DE = GF
\]
\[
4x = 3x + 7
\]
Solving for \( x \):
\[
4x - 3x = 7
\]
\[
x = 7
\]

### Step 3: Find \( DE \)
Now substitute \( x = 7 \) into \( DE = 4x \):
\[
DE = 4(7) = 28 \text{ inches}
\]

### Final Answer:
- \( DE \cong GF \), so \( DE = 28 \) inches.
  
Thus, the correct choices are:
- \( DE \cong GF \)
- \( DE = 28 \)

Would you like further details or explanations? Here are some related questions:

1. What are the properties of parallelograms and how do they apply here?
2. How would you prove two diagonals bisect each other in a parallelogram?
3. If \( EF = 4 \) and \( DG = x + 9 \), how would you solve for \( DG \)?
4. Can you verify if \( P \) is the midpoint of the diagonals?
5. How would the situation change if \( DEFG \) was not a parallelogram?

**Tip**: In a parallelogram, opposite sides are congruent and diagonals bisect each other!

Match the correct answers in the quadrilateral DEFG where diagonals intersect at point P, and sides are labeled with expressions involving x.

Learn how to solve for the side lengths in quadrilateral DEFG, where opposite sides are congruent. The problem involves basic algebra and geometric theorems about parallelograms. Suitable for students in Grades 7-9.

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