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The diagram you have provided contains two parallel lines, \(\overrightarrow{AB}\) and \(\overrightarrow{EH}\), and two transversals, \(\overrightarrow{CF}\) and \(\overrightarrow{GF}\). The angles formed and the given algebraic expressions (for example, \(10x + 20\) and \(12x\)) suggest that this is a problem related to corresponding or alternate angles, likely dealing with the properties of parallel lines and transversals.

### Step 1: Setting up the equation
Because \(\overrightarrow{AB} \parallel \overrightarrow{EH}\), and \(\overrightarrow{CF}\) is a transversal, we can use the fact that angles formed by a transversal cutting through parallel lines are congruent or supplementary (depending on their position). For example, angle \(\angle ABC\) and angle \(\angle GFH\) are either corresponding angles or alternate interior angles, so:

\[
10x + 20 = 12x
\]

### Step 2: Solving for \(x\)
Now, we solve the equation:

\[
10x + 20 = 12x
\]
\[
20 = 12x - 10x
\]
\[
20 = 2x
\]
\[
x = 10
\]

### Step 3: Finding angles \(\angle ABC\) and \(\angle GFH\)
Now that we know \(x = 10\), we can substitute this value into the expressions for the angles.

- For \(\angle ABC\): 
  \[
  10x + 20 = 10(10) + 20 = 100 + 20 = 120^\circ
  \]

- For \(\angle GFH\): 
  \[
  12x = 12(10) = 120^\circ
  \]

### Final Answer:
- Equation: \(10x + 20 = 12x\)
- \(x = 10\)
- \(\angle ABC = 120^\circ\)
- \(\angle GFH = 120^\circ\)

Would you like more details, or do you have any further questions?

### 5 Relative Questions:
1. What are corresponding angles, and why are they congruent?
2. How do alternate interior angles relate to parallel lines?
3. Can supplementary angles also occur with transversals? If so, how?
4. How can you determine if two lines are parallel based on angles?
5. What happens if the transversal forms unequal angles with the parallel lines?

### Tip:
When dealing with parallel lines and transversals, identifying angle relationships like corresponding, alternate interior, and supplementary angles can simplify solving for unknowns.

Given a diagram with parallel lines AB and EH and two transversals CF and GF, find the value of x and calculate the angles ∠ABC and ∠GFH.

Learn how to solve a problem involving parallel lines AB and EH cut by transversals CF and GF. Calculate the value of x and determine the angles ∠ABC and ∠GFH using the properties of corresponding and alternate interior angles. Suitable for students in Grades 8-10.

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