Proof:

Given: ABCD is a square and E, F, and G are the midpoints of sides AB, BC, and CD respectively.

To prove: Triangles AEF and DGF are congruent.

Proof:

1. Draw the square ABCD:
Let's start by drawing the square ABCD.

2. Mark the midpoints:
Next, mark the midpoints E, F, and G on sides AB, BC, and CD respectively.

3. Draw diagonal AC:
Now, draw the diagonal AC of the square.

4. Observe the diagonals:
Since ABCD is a square, the diagonal AC bisects it. Therefore, the point G lies on the diagonal AC.

5. Observe the triangles:
Now, we can observe that triangles AEF and DGF share the side GF, which is common to both triangles.

6. Prove congruence:
To prove that triangles AEF and DGF are congruent, we need to show that the corresponding sides and angles are equal.

a. Side GF:
Since G is the midpoint of side CD and F is the midpoint of side BC, we can conclude that GF is a common side with equal length in both triangles.

b. Side AE:
Since E is the midpoint of side AB, triangle AEF has AE as its base. Similarly, triangle DGF has DG as its base. Since AD is a side of the square ABCD, we can conclude that AE = DG.

c. Side AF:
Triangle AEF has AF as its height, whereas triangle DGF has GF as its height. Since GF is a common side, we can conclude that AF = GF.

d. Angle AFE and angle DFG:
Both angles are right angles since they are formed by a side of the square and a diagonal. Therefore, angle AFE = angle DFG.

7. Conclusion:
Since triangles AEF and DGF have equal corresponding sides and angles, we can conclude that they are congruent by the Side-Angle-Side (SAS) congruence criterion.

Therefore, triangles AEF and DGF are congruent.