This diagram seems to represent angles in a geometric figure with multiple lines intersecting at point $G$. You are asked to find the measures of $\angle AGB$ and $\angle CGB$.

Observations:

1. The diagram has several lines intersecting at $G$, and the label “32” likely represents the measure of $\angle DGC$, which forms a smaller angle within the figure.
2. Line $AB$ and line $CE$ appear to form right angles, as shown with the square symbol near $G$.

Step-by-Step Solution:

1. Identify known angles:

   • $\angle DGC = 32^\circ$ (given in the diagram).

2. Use properties of straight angles:

   • Since line $AB$ and line $CD$ form a straight line, $\angle DGB + \angle CGB = 180^\circ$.
   • From this, $\angle CGB = 180^\circ - 32^\circ = 148^\circ$.

3. Use vertical angle property:

   • $\angle AGB$ and $\angle CGB$ are vertical angles, so they are equal.
   • Therefore, $\angle AGB = \angle CGB = 148^\circ$.

Thus, both $m\angle AGB$ and $m\angle CGB$ are $148^\circ$.

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

Here are five related questions for further practice:

1. What is the measure of $\angle DGB$?
2. How do you find the complementary angle to $\angle DGC$?
3. What are the properties of vertical angles?
4. How do supplementary angles work in geometric configurations?
5. What other types of angles can you form when lines intersect?

Tip: Remember that vertical angles are always congruent when two lines intersect!