The given figure represents a parallelogram with the following details:

1. $FD = 22$
2. $CG = 7$ and $GD$ is unknown.
3. $CD = 15$ and $DE = 10$.

We aim to calculate the lengths of $CE$ and $GD$.

Step 1: Use diagonals of parallelogram property

In any parallelogram:

• The diagonals bisect each other. This means: $FG = GD$ and: $CG + GD = CD.$ Substituting the known values: $7 + GD = 15.$ Solving: $GD = 8.$

Step 2: Calculate $CE$

Since $CE$ is a diagonal of the parallelogram, it is divided equally at point $G$. Hence: $CE = 2 \times GE.$ From the diagram: $GE = DE = 10.$ So: $CE = 2 \times 10 = 20.$

Final Results:

1. $GD = 8$.
2. $CE = 20$.

Would you like a step-by-step explanation or more examples of similar problems? Here are some additional questions for deeper understanding:

1. What are the properties of diagonals in different types of quadrilaterals?
2. How do we find the area of a parallelogram given sides and angles?
3. Can the lengths of diagonals help us identify specific types of parallelograms?
4. How would this solution change if the figure were not a parallelogram?
5. What other properties can be derived from symmetry in a parallelogram?

Tip: Always remember that the diagonals of a parallelogram bisect each other, a key property in geometry!