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.