https://file.aiquickdraw.com/mathbot/file/1726622274616f8u7f8y0.jpg

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.

Solve the angle problems provided in a worksheet image, identifying angle relationships and solving for missing variables.

Learn how to solve geometry problems involving vertical, supplementary, and corresponding angles. Includes step-by-step solutions to find unknown variables, suitable for Grades 8-10.

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