Let's break down the tasks based on the image you uploaded:

Part A:

For each diagram, the instructions ask to:

1. State the type of angles given.
2. State their relationship.
3. Find the value of $x$.

Here’s a step-by-step approach to solve the problems:

1. Diagram 1:

• Type of angles: Vertical angles (opposite angles when two lines intersect).
• Relationship: Vertical angles are equal.
• Equation: $80^\circ = 2x$, solve for $x$: $2x = 80 \implies x = 40$

2. Diagram 2:

• Type of angles: Vertical angles.
• Relationship: Vertical angles are equal.
• Equation: $120^\circ = 3x$, solve for $x$: $3x = 120 \implies x = 40$

3. Diagram 3:

• Type of angles: Supplementary angles (sum to $180^\circ$).
• Relationship: The two angles are on a straight line, so they add up to $180^\circ$.
• Equation: $2x + 110 = 180$, solve for $x$: $2x = 180 - 110 \implies 2x = 70 \implies x = 35$

4. Diagram 4:

• Type of angles: Corresponding angles.
• Relationship: Corresponding angles are equal.
• Equation: $80^\circ = x + 15$, solve for $x$: $x + 15 = 80 \implies x = 80 - 15 \implies x = 65$

5. Diagram 5:

• Type of angles: Vertical angles.
• Relationship: Vertical angles are equal.
• Equation: $68^\circ = 2x$, solve for $x$: $2x = 68 \implies x = 34$

6. Diagram 6:

• Type of angles: Supplementary angles.
• Relationship: The two angles are on a straight line, so they add up to $180^\circ$.
• Equation: $92^\circ + (2x - 4) = 180$, solve for $x$: $2x - 4 = 180 - 92 \implies 2x - 4 = 88 \implies 2x = 92 \implies x = 46$

Part B:

Find the missing variables based on the relationships of the angles.

7. Diagram 7:

• Type of angles: Corresponding angles.
• Equation 1: $75^\circ = 5x - 10$, solve for $x$: $5x - 10 = 75 \implies 5x = 85 \implies x = 17$
• Equation 2: $75^\circ = 10y - 25$, solve for $y$: $10y - 25 = 75 \implies 10y = 100 \implies y = 10$

8. Diagram 8:

• Type of angles: Alternate interior angles.
• Equation 1: $60^\circ = 3x + 15$, solve for $x$: $3x + 15 = 60 \implies 3x = 45 \implies x = 15$
• Equation 2: $60^\circ = 5y - 10$, solve for $y$: $5y - 10 = 60 \implies 5y = 70 \implies y = 14$

9. Diagram 9:

• Type of angles: Alternate interior angles.
• Equation: $102^\circ = 2x + 12$, solve for $x$: $2x + 12 = 102 \implies 2x = 90 \implies x = 45$

Would you like more detailed explanations for any of these steps, or have any questions?

Here are five additional practice questions to expand on this:

1. What is the relationship between alternate exterior angles?
2. How do you identify corresponding angles in a pair of parallel lines?
3. Solve for $x$ if the supplementary angle to $70^\circ$ is given as $2x + 10$.
4. How are vertical angles and adjacent angles different?
5. Can two acute angles ever be supplementary? Why or why not?

Tip: Always remember that angles on a straight line sum up to $180^\circ$ (linear pair).