To solve this problem, let's follow these steps:

1. Understand the figure and given information:

   • ABCD is a trapezoid with $AD \parallel BC$.
   • E, F, and G are midpoints of AD, BC, and DC respectively.
   • The areas of the triangles $\triangle ABE$, $\triangle EGD$, and $\triangle GFC$ are provided as 2, 5, and 6 respectively.

2. Determine the method:

   • We'll use the areas given and properties of midpoints to find the area of the shaded region $\triangle EGF$.

3. Calculations:

Let's denote the area of the trapezoid ABCD as $A_{ABCD}$.

• Since E and F are midpoints, $EF$ is parallel to $AD$ and $BC$, and $EF$ is the midsegment of the trapezoid.

Using the properties of midsegments, the area of $\triangle EGF$ can be calculated as follows:

• The shaded area $\triangle EGF$ is composed of parts of triangles formed by connecting midpoints.

• The total area of triangles $\triangle ABE$, $\triangle EGD$, and $\triangle GFC$ plus the area of the shaded region $\triangle EGF$ should equal the total area of the trapezoid $ABCD$.

Given: $\text{Area of } \triangle ABE = 2$ $\text{Area of } \triangle EGD = 5$ $\text{Area of } \triangle GFC = 6$

Let's denote the area of the shaded region $\triangle EGF$ as $A_{EGF}$.

The total area of the trapezoid ABCD: $A_{ABCD} = 2 + 5 + 6 + A_{EGF}$

Notice that the trapezoid is divided into 4 regions. The area of the trapezoid is twice the sum of the areas of these triangles because E, F, and G are midpoints: $A_{ABCD} = 2 \times (2 + 5 + 6)$

Thus: $2 \times (2 + 5 + 6) = 2 \times 13 = 26$

So, the equation is: $26 = 2 + 5 + 6 + A_{EGF}$

$26 = 13 + A_{EGF}$

$A_{EGF} = 26 - 13 = 13$

Therefore, the area of the shaded region $\triangle EGF$ is $\boxed{13}$.

Would you like further details or have any questions about this solution?

Here are some questions you might ask next:

1. How do midpoints affect the division of areas in geometric shapes?
2. What other properties of trapezoids can be useful in area calculations?
3. Can this method be applied to other types of quadrilaterals?
4. What role do parallel lines play in area determination?
5. Are there other methods to find the area of the shaded region?

Tip: When solving geometric problems, always look for symmetrical properties and relationships between midpoints and parallel lines.