Consider the following figure.

A line and two segments form five labeled angles.

The line is horizontal.

The first segment starts near the left side of the line, travels up and to the left, and ends at an unknown distance.

The second segment starts near the right side of the line, travels up and to the left, and ends where the first segment ends.

∠1 is formed by the bottom left side of the first segment and the top side of the line.

∠2 is formed by the bottom right side of the first segment and the top side of the line.

∠3 is formed by the bottom left side of the second segment and the top side of the line.

∠4 is formed by the bottom right side of the second segment and the top side of the line.

∠5 is formed by the top right side of the first segment and the top left side of the second segment.

Given:

m∠1 = (8(*x* + 2))°

m∠3 = (8*x* − 6)°

m∠5 = (8(*x* + 1) − 2)°

Find:*x*

*x* =  

To solve for \( x \), we will use the relationships between the given angles and the fact that they add up to certain values due to their geometrical arrangement.

### Step 1: Analyze the geometry of the figure.
From the description, we can make the following observations:

- Since the line is horizontal, it forms a straight line, meaning the sum of the angles on one side of this line must be \( 180^\circ \).
- \(\angle 1\) and \(\angle 2\) are adjacent angles that form a straight line, so their sum is \( 180^\circ \).
- Similarly, \(\angle 3\) and \(\angle 4\) form a straight line, so their sum is also \( 180^\circ \).
- \(\angle 5\) is the angle between the two segments that meet at a common vertex.

### Step 2: Write down the known values.

- \( m\angle 1 = 8(x + 2)^\circ \)
- \( m\angle 3 = 8x - 6^\circ \)
- \( m\angle 5 = 8(x + 1) - 2^\circ \)

### Step 3: Use the fact that \( \angle 1 \) and \( \angle 3 \) form adjacent angles on opposite sides of the horizontal line.

Since the line is horizontal, the angles on either side should sum up to \( 180^\circ \):
\[
m\angle 1 + m\angle 3 = 180^\circ
\]
Substitute the expressions for \( m\angle 1 \) and \( m\angle 3 \):
\[
8(x + 2) + (8x - 6) = 180
\]
Simplify the equation:
\[
8x + 16 + 8x - 6 = 180
\]
\[
16x + 10 = 180
\]
Subtract 10 from both sides:
\[
16x = 170
\]
Now, divide by 16:
\[
x = \frac{170}{16} = 10.625
\]

Thus, the value of \( x \) is \( 10.625 \).

Would you like further details on the steps or more clarification? Here are some related questions:

1. How do complementary and supplementary angles differ?
2. Can we apply the same angle rules to parallel lines cut by a transversal?
3. What is the significance of angle bisectors in geometry?
4. How are angles categorized in geometry (e.g., acute, obtuse)?
5. How can we use algebra to find unknown angle measures in more complex polygons?

**Tip:** Always consider the geometric relationships of angles (e.g., complementary, supplementary) when solving angle problems.

Consider the following figure. A line and two segments form five labeled angles. The line is horizontal. The first segment starts near the left side of the line, travels up and to the left, and ends at an unknown distance. The second segment starts near the right side of the line, travels up and to the left, and ends where the first segment ends. ∠1 is formed by the bottom left side of the first segment and the top side of the line. ∠2 is formed by the bottom right side of the first segment and the top side of the line. ∠3 is formed by the bottom left side of the second segment and the top side of the line. ∠4 is formed by the bottom right side of the second segment and the top side of the line. ∠5 is formed by the top right side of the first segment and the top left side of the second segment. Given: m∠1 = (8(x + 2))°, m∠3 = (8x − 6)°, m∠5 = (8(x + 1) − 2)°. Find: x

Learn how to solve for x using supplementary angles in a geometric figure involving a line and two segments. This problem explores key concepts in algebra and geometry, applying the supplementary angles and linear pair theorems. Suitable for students in Grades 9-12.

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