https://file.aiquickdraw.com/mathbot/file/1726864939016jgwi8aco.jpg

Let's analyze the given diagram and solve for the measure of \( \angle 6 \).

### Step-by-step Solution:
- The diagram shows two parallel lines intersected by a transversal.
- From the image, we know:
  - \( \angle 5 = 45^\circ \) (given),
  - \( \angle 2 = 5x + 35^\circ \),
  - \( \angle 3 \) and \( \angle 5 \) are alternate interior angles, and alternate interior angles are congruent when the lines are parallel.
  Therefore, \( \angle 3 = \angle 5 = 45^\circ \).
  
Since \( \angle 3 \) and \( \angle 2 \) form a linear pair, we know:
\[
\angle 3 + \angle 2 = 180^\circ
\]
Substitute the values:
\[
45^\circ + (5x + 35^\circ) = 180^\circ
\]
Simplifying the equation:
\[
45^\circ + 5x + 35^\circ = 180^\circ
\]
\[
80^\circ + 5x = 180^\circ
\]
\[
5x = 180^\circ - 80^\circ
\]
\[
5x = 100^\circ
\]
\[
x = 20^\circ
\]

### Now that we know \( x = 20^\circ \), we can find \( \angle 2 \):
\[
\angle 2 = 5x + 35^\circ = 5(20^\circ) + 35^\circ = 100^\circ + 35^\circ = 135^\circ
\]

### Finally, let's determine \( \angle 6 \):
- \( \angle 6 \) and \( \angle 2 \) are corresponding angles, and corresponding angles are congruent when lines are parallel.
Thus, \( \angle 6 = \angle 2 = 135^\circ \).

### Answer:
The measure of \( \angle 6 \) is **135°**.

Would you like a detailed explanation of any specific part of the solution? 

Here are some related questions for further practice:
1. What is the measure of \( \angle 8 \) based on the diagram?
2. How would you find \( \angle 4 \) using the given information?
3. What is the sum of all the interior angles formed by the transversal?
4. Can you explain how alternate interior angles work in this diagram?
5. If the measure of \( \angle 3 \) were different, how would that affect \( \angle 6 \)?

**Tip**: When dealing with parallel lines and a transversal, remember that corresponding angles and alternate interior angles are always congruent!

Learn how to find the measure of \( \angle 6 \) in a diagram involving parallel lines and a transversal. This problem uses the Corresponding Angles Postulate and Alternate Interior Angles Theorem to solve for the unknown angle. Suitable for students in Grades 7-10.

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