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the answer for 30 is: Vertical Angles, x=4, 17 degrees, and 163 degrees. the answer for 31 is angle CGE is 42 degrees

Let's go through each problem step-by-step:

### Problem 30:
We are given two intersecting lines, and we need to identify the name of the angle pair, solve for \(x\), and determine the measures of all four angles in the diagram.

#### Step 1: Identifying the angle pair
The two given angles form **Vertical Angles** because they are opposite each other when two lines intersect.

#### Step 2: Solving for \(x\)
The expressions for the angles are:

\[
3x + 5 \quad \text{and} \quad 8x - 15
\]

Since they are vertical angles, they are congruent (equal). Therefore, we set the expressions equal to each other:

\[
3x + 5 = 8x - 15
\]

Now, solve for \(x\):

1. Subtract \(3x\) from both sides:
   \[
   5 = 5x - 15
   \]
2. Add 15 to both sides:
   \[
   20 = 5x
   \]
3. Divide both sides by 5:
   \[
   x = 4
   \]

#### Step 3: Finding the angle measures
Now that we have \(x = 4\), substitute it back into the expressions for the angles:

1. \(3x + 5 = 3(4) + 5 = 12 + 5 = 17^\circ\)
2. \(8x - 15 = 8(4) - 15 = 32 - 15 = 17^\circ\)

Since vertical angles are equal, we have two pairs of equal angles: \(17^\circ\) and its vertical angle is \(17^\circ\). The adjacent angles are supplementary (add up to 180°), so:

\[
180^\circ - 17^\circ = 163^\circ
\]

Thus, the four angles are:
- \(17^\circ\),
- \(17^\circ\),
- \(163^\circ\),
- \(163^\circ\).

### Problem 31:
We are given that \(AD \parallel BE\) and that angle \( \angle BCF = 138^\circ \). We need to find the measure of \( \angle CGE \).

Since \(AD \parallel BE\), the angles form **corresponding angles** or **co-interior angles**.

#### Step 1: Using Co-interior (or supplementary) angles
Angle \(BCF\) and \(CGE\) are on the same side of the transversal, and thus they are **co-interior angles**. Co-interior angles are supplementary, meaning their sum is 180°:

\[
\angle BCF + \angle CGE = 180^\circ
\]

Substitute \( \angle BCF = 138^\circ \):

\[
138^\circ + \angle CGE = 180^\circ
\]

Now, solve for \( \angle CGE \):

\[
\angle CGE = 180^\circ - 138^\circ = 42^\circ
\]

Thus, \( \angle CGE = 42^\circ \).

---

### Summary:
- For **Problem 30**, the angle pair is **Vertical Angles**, \(x = 4\), and the four angle measures are \(17^\circ\) and \(163^\circ\).
- For **Problem 31**, \( \angle CGE = 42^\circ \).

---

Do you have any questions or would you like more details?

Here are 5 related questions to explore further:
1. What are alternate interior angles, and how do they differ from corresponding angles?
2. How would the solution change if \(3x + 5\) and \(8x - 15\) were supplementary instead of vertical angles?
3. Can you explain how to identify linear pairs of angles in intersecting lines?
4. What other pairs of angles exist when two lines intersect, apart from vertical angles?
5. How can you use algebra to find unknown angles when more than two expressions are involved?

**Tip:** Always remember that vertical angles are congruent and that supplementary angles add up to 180°. This can help you solve for unknown angles efficiently.

Learn how to solve geometry problems involving vertical and supplementary angles with detailed step-by-step solutions. This resource covers how to determine the values of angles in intersecting lines and parallel line scenarios, appropriate for high school level studies.

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