To solve for the values of $x$, $y$, and $z$, let us analyze the diagram and use the following geometric rules:

1. Alternate Interior Angles Theorem: Since $CE \parallel AB$ and $CD \parallel BG$, alternate interior angles formed by the transversal lines are equal.

2. Sum of Angles Around a Point: The sum of angles at a point is $360^\circ$.

3. Linear Pair Angles: Adjacent angles on a straight line sum up to $180^\circ$.

Step-by-Step Solution

1. Angle $z$:

   • From the figure, the sum of angles around point $C$ is $360^\circ$.
   • Angles at $C$ are $88^\circ + 15^\circ + z$.
   • Equation: $88^\circ + 15^\circ + z = 180^\circ$
   • Simplify: $z = 180^\circ - (88^\circ + 15^\circ)$ $z = 180^\circ - 103^\circ = 77^\circ$

2. Angle $x$:

   • Angle $x$ corresponds to the alternate interior angle of $z$, since $CE \parallel AB$ and $CD$ is the transversal.
   • Hence: $x = z = 77^\circ$

3. Angle $y$:

   • Using the linear pair property, $y$ and $x$ form a straight line. Therefore: $x + y = 180^\circ$
   • Substituting $x = 77^\circ$: $77^\circ + y = 180^\circ$ $y = 180^\circ - 77^\circ = 103^\circ$

Final Values:

• $x = 77^\circ$
• $y = 103^\circ$
• $z = 77^\circ$

Let me know if you need a detailed explanation or have further questions! Here are some related questions to expand understanding:

1. Why do alternate interior angles form equal measures when lines are parallel?
2. How is the sum of angles around a point derived geometrically?
3. What other properties of parallel lines can be applied to such problems?
4. Can we solve for $y$ first instead of $z$? How would it affect the steps?
5. How can this problem be modified if $CE$ or $CD$ were not parallel to other lines?

Tip: Always identify all parallel lines and their transversals when solving geometry problems involving angles.