Let's analyze the diagram and solve the questions step-by-step.

1. Measure of Angle 8:

   • From the diagram, angle 8 is vertically opposite to the $45^\circ$ angle, which makes them equal. Therefore: $\text{Angle 8} = 45^\circ$

2. Measure of Angle 3:

   • Angle 3 and the angle labeled as $5x + 35^\circ$ form a pair of corresponding angles, as they are located on parallel lines and are both on the same side of the transversal.
   • Therefore, $\text{Angle 3} = 5x + 35^\circ$.

3. Value of $x$:

   • Angle 6 and angle 3 are alternate interior angles, so they must be equal. Since angle 6 is $45^\circ$, we set the equation: $5x + 35^\circ = 45^\circ$ Solving for $x$: $5x = 45^\circ - 35^\circ$ $5x = 10^\circ$ $x = 2$

Final Answers:

1. Measure of angle 8: $45^\circ$
2. Measure of angle 3: $5x + 35^\circ = 45^\circ$
3. Value of $x = 2$

Would you like more details on any of these steps?

Here are 5 follow-up questions to extend the problem:

1. What is the measure of angle 2?
2. Can you explain the relationship between angles 4 and 8?
3. How can you verify that lines in the diagram are parallel using angles?
4. What is the sum of angles around a point like angle 1 and angle 8?
5. How do corresponding angles help in solving for unknown variables in such diagrams?

Tip: When two parallel lines are cut by a transversal, corresponding angles are equal, and alternate interior angles are also equal!