Let's go through the questions shown in the image.

Part A

For each of the given diagrams, we are asked to:

1. State the type of angles (e.g., complementary, supplementary, vertical, corresponding).
2. State their relationship (e.g., are the angles equal, supplementary, etc.).
3. Find the value of $x$.

Let’s address each problem step-by-step:

1.

• Type of angles: The angles are vertically opposite angles.
• Relationship: Vertically opposite angles are equal.
• Equation: $2x = 80^\circ$ $x = \frac{80^\circ}{2} = 40^\circ$

2.

• Type of angles: The angles are supplementary (they form a straight line).
• Relationship: The sum of supplementary angles is $180^\circ$.
• Equation: $120^\circ + 3x = 180^\circ$ $3x = 180^\circ - 120^\circ = 60^\circ$ $x = \frac{60^\circ}{3} = 20^\circ$

3.

• Type of angles: The angles are supplementary (linear pair).
• Relationship: The sum is $180^\circ$.
• Equation: $2x + 110^\circ = 180^\circ$ $2x = 180^\circ - 110^\circ = 70^\circ$ $x = \frac{70^\circ}{2} = 35^\circ$

4.

• Type of angles: The angles are vertically opposite angles.
• Relationship: Vertically opposite angles are equal.
• Equation: $x + 15^\circ = 80^\circ$ $x = 80^\circ - 15^\circ = 65^\circ$

5.

• Type of angles: The angles are vertically opposite angles.
• Relationship: Vertically opposite angles are equal.
• Equation: $68^\circ = 2x$ $x = \frac{68^\circ}{2} = 34^\circ$

6.

• Type of angles: The angles are vertically opposite angles.
• Relationship: Vertically opposite angles are equal.
• Equation: $92^\circ = 2x - 4^\circ$ $2x = 92^\circ + 4^\circ = 96^\circ$ $x = \frac{96^\circ}{2} = 48^\circ$

Part B: Find the missing variables

7.

• Type of angles: The angles are vertically opposite angles.
• Relationship: Vertically opposite angles are equal.
• Equation: $10y - 25 = 75^\circ$ $10y = 75^\circ + 25^\circ = 100^\circ$ $y = \frac{100^\circ}{10} = 10^\circ$

8.

• Type of angles: The angles are corresponding angles (parallel lines).

• Relationship: Corresponding angles are equal.

• Equation: $3x + 15^\circ = 60^\circ$ $3x = 60^\circ - 15^\circ = 45^\circ$ $x = \frac{45^\circ}{3} = 15^\circ$

  And for $y$, the equation is $5y - 10^\circ = 60^\circ$. $5y = 60^\circ + 10^\circ = 70^\circ$ $y = \frac{70^\circ}{5} = 14^\circ$

9.

• Type of angles: The angles are vertically opposite angles.
• Relationship: Vertically opposite angles are equal.
• Equation: $2x + 12^\circ = 102^\circ$ $2x = 102^\circ - 12^\circ = 90^\circ$ $x = \frac{90^\circ}{2} = 45^\circ$

Would you like more details or have any questions? Here are some related questions for practice:

1. What are complementary angles, and how do they differ from supplementary angles?
2. How can you identify vertically opposite angles in a diagram?
3. In a diagram with parallel lines, how do you determine corresponding angles?
4. What is the sum of the interior angles of a triangle, and how does that help solve angle problems?
5. How can the exterior angle theorem be applied to find unknown angles in triangles?

Tip: Always check if the angles are forming straight lines (supplementary) or intersecting lines (vertical). It helps to quickly identify relationships.