https://file.aiquickdraw.com/mathbot/file/1725659354214n8zq7tty.jpg

The image shows a geometry problem involving a rectangle \(ABCD\) and additional line segments inside it.

Let's analyze the problem step-by-step:

1. **Given Information**:
   - \(ABCD\) is a rectangle.
   - \(DE = CG\).
   - Triangles \(\triangle ADE\) and \(\triangle BCG\) are mentioned.

2. **Part (a)**: Prove that \(\triangle ADE \cong \triangle BCG\).

   To prove that \(\triangle ADE \cong \triangle BCG\), we can use the given information:
   - \(DE = CG\) (given).
   - \(AB = CD\) and \(AD = BC\) since \(ABCD\) is a rectangle.
   - \(\angle ADE = \angle BCG = 90^\circ\) since \(ABCD\) is a rectangle, and both are right triangles.
   
   By the Side-Angle-Side (SAS) criterion, \(\triangle ADE \cong \triangle BCG\).

3. **Part (b)**: List the three corresponding congruent elements.

   - \(AD = BC\)
   - \(DE = CG\)
   - \(\angle ADE = \angle BCG = 90^\circ\)

4. **Part (c)**: Given \(AE = 5\), \(EG = 2\), \(DE = 4\), calculate the length of \(AB\).

   - Since \(\triangle ADE \cong \triangle BCG\), we have \(DE = CG = 4\).
   - Also, \(AB = AD + DE + CG = AE + EG + DE = 5 + 2 + 4 = 11\).

5. **Calculate the Perimeter of Quadrilateral \(ABGE\)**:

   - \(AB = 11\)
   - \(BG = BC - CG = AB - DE = 11 - 4 = 7\)
   - \(AE = 5\)
   - \(EG = 2\)

   Therefore, the perimeter of \(ABGE\) is:
   \[
   AB + BG + GE + AE = 11 + 7 + 2 + 5 = 25.
   \]

Would you like a more detailed explanation or have any questions?

### Additional Related Questions:
1. What are the criteria for triangle congruence?
2. How can you calculate the area of rectangle \(ABCD\)?
3. What is the formula for the perimeter of any quadrilateral?
4. How to determine if two triangles are similar rather than congruent?
5. What is the difference between congruence and similarity in geometry?

### Tip:
For geometry problems, always look for parallel lines, congruent segments, and angle properties to find relationships between different parts of the shapes.

In rectangle ABCD, prove that triangles ADE and BCG are congruent. Given that DE = CG, and AE = 5, EG = 2, DE = 4, calculate the length of AB and the perimeter of quadrilateral ABGE.

Solve a geometry problem involving rectangle ABCD, where you prove triangle congruence using the SAS criterion and calculate the perimeter of quadrilateral ABGE. The problem includes properties of rectangles and congruent triangles. Suitable for students in Grades 7-9.

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