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In this problem, we are dealing with parallel lines cut by a transversal. The angles provided, \(109^\circ\) and \(3x + 10^\circ\), are alternate interior angles. For parallel lines, alternate interior angles are equal.

Thus, we can set up the following equation:

\[
3x + 10 = 109
\]

### Solving for \(x\):
\[
3x = 109 - 10
\]
\[
3x = 99
\]
\[
x = \frac{99}{3} = 33
\]

Thus, the value of \(x\) is \(33\).

### Finding the angle:
Now substitute \(x = 33\) back into the expression for the angle:

\[
3x + 10 = 3(33) + 10 = 99 + 10 = 109^\circ
\]

So, the angle measure is \(109^\circ\).

Would you like more details or have any questions? Here are five related questions:

1. What are corresponding angles in parallel lines and a transversal?
2. How would you find the value of \(x\) if the lines were not parallel?
3. What is the relationship between supplementary angles in parallel lines?
4. How do you identify alternate exterior angles?
5. What happens when two parallel lines are cut by more than one transversal?

**Tip**: In geometry, recognizing angle relationships in parallel lines can simplify solving for unknown variables quickly.

Find the value of x and the angle measure.

Learn how to solve for x and find the angle measure using the Alternate Interior Angle Theorem. This problem involves parallel lines cut by a transversal, where the angles are equal. Suitable for students in Grades 8-10.

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