https://file.aiquickdraw.com/mathbot/file/1726251253347fwry2ws9.jpg

Let's go through the image step by step and solve the problems.

### 1. Problem 2: Calculate the value of \( t \)
You have the equation:
\[
3t + 12 = 66
\]
Steps:
- Subtract 12 from both sides:
  \[
  3t = 66 - 12 = 54
  \]
- Divide both sides by 3:
  \[
  t = \frac{54}{3} = 18
  \]
So, the value of \( t \) is **18**.

### 2. Problem 3: Calculate the value of \( p \)
The angles form a linear pair, meaning their sum is \(180^\circ\). Given:
\[
2p + 30^\circ \quad \text{and} \quad 108^\circ
\]
You can set up the equation:
\[
2p + 30 + 108 = 180
\]
Steps:
- Combine like terms:
  \[
  2p + 138 = 180
  \]
- Subtract 138 from both sides:
  \[
  2p = 180 - 138 = 42
  \]
- Divide both sides by 2:
  \[
  p = \frac{42}{2} = 21
  \]
So, the value of \( p \) is **21**.

### 3. Problem 4: Calculate the value of \( z \)
This problem involves vertical angles, which are equal. Therefore:
\[
58^\circ = 2z - 10^\circ
\]
Steps:
- Add 10 to both sides:
  \[
  58 + 10 = 2z \quad \Rightarrow \quad 68 = 2z
  \]
- Divide both sides by 2:
  \[
  z = \frac{68}{2} = 34
  \]
So, the value of \( z \) is **34**.

---

Would you like more details on any of the steps? Here are five related questions to explore further:

1. What are the properties of vertical angles and how can they help solve similar problems?
2. How do you determine when angles form a linear pair and why their sum is \(180^\circ\)?
3. What happens to an equation when both sides are divided by a constant?
4. How would you solve problems with more complex algebraic expressions involving angles?
5. What are some real-world applications of linear and vertical angle concepts?

**Tip:** Always check whether angles are complementary (sum to \(90^\circ\)) or supplementary (sum to \(180^\circ\)) to guide your problem-solving approach.

The image contains algebraic problems where the user is asked to calculate the values of t, p, and z based on given equations involving angles.

Learn how to solve algebraic problems involving angles with detailed step-by-step solutions. This problem covers key concepts such as linear equations, supplementary angles, and vertical angles, suitable for Grades 6-8 students.

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