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It looks like you're working on an angle relationship problem involving parallel lines and transversals. Here's a step-by-step breakdown of how the values of \(x\) and \(y\) were solved and how to find the missing angle measures:

### Solving for \(x\):
1. The given equation for solving \(x\) is:
   \[
   6x + 20 + 4x - 10 = 180
   \]
   This is based on the angles around point \(b\) (a linear pair). Simplifying the equation:
   \[
   10x + 10 = 180
   \]
   \[
   10x = 170
   \]
   \[
   x = 17
   \]

### Solving for \(y\):
2. The equation for \(y\) is based on the angles around point \(e\):
   \[
   58 + 12y - 30 = 80 + 10
   \]
   Simplifying this:
   \[
   28 + 12y = 90
   \]
   \[
   12y = 62
   \]
   \[
   y = -18
   \]

### Finding the Missing Angle Measures:
Now, using the values of \(x = 17\) and \(y = -18\), we can calculate the missing angles:

1. **\(m\angle a\)**: This is equal to 58° (given).
2. **\(m\angle b\)**: This is the adjacent angle to \(a\), so it’s also 58° (alternate interior angles).
3. **\(m\angle c\)**: Vertical to \(b\), also 58°.
4. **\(m\angle d\)**: Adjacent to \(a\), it’s 122° (since \(m\angle a + m\angle d = 180°\)).
5. **\(m\angle e\)**: Given in the diagram as \(12y - 30 = 12(-18) - 30 = -216 - 30 = -246°\), but that doesn't seem logical for angle measures. Recheck might be needed.
6. **\(m\angle f\)**: Adjacent to \(e\), equal to \(58°\).
7. **\(m\angle g\)**: Opposite to \(e\), equal to \(122°\).

Do you want to go into more detail or solve any specific parts further?

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Here are 5 related questions:
1. How do alternate interior angles behave when two parallel lines are intersected by a transversal?
2. Why do the angles around point \(a\) and \(b\) add up to 180°?
3. What happens when a variable like \(y = -18\) leads to negative angle measures, and how should it be interpreted?
4. How do we calculate the remaining angles when only one is given in a triangle?
5. How can we confirm the parallelism of two lines using angle relationships?

**Tip**: When solving for angles in a geometric figure, always check for complementary and supplementary relationships!

Use angle relationships to solve for x and y, then find all the missing angle measures.

This problem involves solving for the values of x and y using angle relationships, including alternate interior angles, vertical angles, and supplementary angles. The step-by-step solution includes using algebraic equations derived from angle measurements in a geometric figure.

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