https://file.aiquickdraw.com/mathbot/file/1727303730939i8bgupif.jpg

The image shows a problem involving two lines cut by a transversal, and the question asks for the value of \( x \) that ensures the lines are parallel.

From the diagram and the solution provided, the angles formed are:

- \( (2x + 4)^\circ \) for one of the angles,
- \( (5x + 8)^\circ \) for the other angle.

Since the lines need to be parallel, these angles are supplementary. This means their sum should be \( 180^\circ \). The equation would be:

\[
(2x + 4) + (5x + 8) = 180
\]

### Step-by-step solution:

1. Combine like terms:
\[
2x + 5x + 4 + 8 = 180
\]
\[
7x + 12 = 180
\]

2. Subtract 12 from both sides:
\[
7x = 180 - 12
\]
\[
7x = 168
\]

3. Divide both sides by 7:
\[
x = \frac{168}{7} = 24
\]

However, the worked-out solution in the image solves the two angles individually.

- For \( (2x + 4)^\circ \), the student first solves \( 2x + 4 = 180 \), which gives \( x = 88 \).
- For \( (5x + 8)^\circ \), the student solves \( 5x + 8 = 180 \), which gives \( x = 34.4 \).

These values indicate two separate steps, possibly due to parallel checks for specific angles, but the actual solution for parallel lines would still involve \( x = 24 \) when equating the sum to \( 180^\circ \).

Let me know if you would like any further clarification or details!

### Follow-up Questions:
1. What is the significance of supplementary angles in parallel lines?
2. How are alternate interior angles related to parallel lines?
3. What other methods can confirm parallelism in geometry?
4. Why do we equate the sum of angles to \( 180^\circ \) in this scenario?
5. Could the values of \( x = 88 \) or \( x = 34.4 \) be used in a different type of problem?

### Tip:
Always double-check the relationship between angles, especially if you are dealing with parallel lines and transversals, as the type of angles (alternate, corresponding, or consecutive) will dictate your approach.

The figure shows two lines cut by a transversal. What value of x shows that the lines are parallel?

This geometry problem asks for the value of x that makes two lines parallel when they are cut by a transversal. The solution involves the concept of supplementary angles and solving algebraic equations.

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