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?

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!