Practice worksheet featuring eight problems where students must use the linear pair theorem to set up equations and solve for the variable x.

To solve the problems using the Linear Pair Theorem, we need to remember that the angles forming a linear pair are supplementary. This means their measures add up to \(180^\circ\).

Let's solve each problem step by step.

---

Problem 1:

\[ 2x + 5 \quad \text{and} \quad 65^\circ \]

#### Step 1: Set up the equation
Since these angles form a linear pair:
\[ (2x + 5) + 65 = 180 \]

#### Step 2: Simplify the equation
\[ 2x + 5 + 65 = 180 \]
\[ 2x + 70 = 180 \]

#### Step 3: Solve for \(x\)
\[ 2x = 180 - 70 \]
\[ 2x = 110 \]
\[ x = \frac{110}{2} \]
\[ x = 55 \]

#### Answer:
\[ x = 55 \]

---

Problem 2:

\[ 136^\circ \quad \text{and} \quad 4x + 12 \]

#### Step 1: Set up the equation
Since these angles form a linear pair:
\[ 136 + (4x + 12) = 180 \]

#### Step 2: Simplify the equation
\[ 136 + 4x + 12 = 180 \]
\[ 4x + 148 = 180 \]

#### Step 3: Solve for \(x\)
\[ 4x = 180 - 148 \]
\[ 4x = 32 \]
\[ x = \frac{32}{4} \]
\[ x = 8 \]

#### Answer:
\[ x = 8 \]

---

Problem 3:

\[ 45^\circ \quad \text{and} \quad 5x + 25 \]

#### Step 1: Set up the equation
Since these angles form a linear pair:
\[ 45 + (5x + 25) = 180 \]

#### Step 2: Simplify the equation
\[ 45 + 5x + 25 = 180 \]
\[ 5x + 70 = 180 \]

#### Step 3: Solve for \(x\)
\[ 5x = 180 - 70 \]
\[ 5x = 110 \]
\[ x = \frac{110}{5} \]
\[ x = 22 \]

#### Answer:
\[ x = 22 \]

---

Problem 4:

\[ 7x - 8 \quad \text{and} \quad 76^\circ \]

#### Step 1: Set up the equation
Since these angles form a linear pair:
\[ (7x - 8) + 76 = 180 \]

#### Step 2: Simplify the equation
\[ 7x - 8 + 76 = 180 \]
\[ 7x + 68 = 180 \]

#### Step 3: Solve for \(x\)
\[ 7x = 180 - 68 \]
\[ 7x = 112 \]
\[ x = \frac{112}{7} \]
\[ x = 16 \]

#### Answer:
\[ x = 16 \]

---

Problem 5:

\[ 3x - 17 \quad \text{and} \quad 122^\circ \]

#### Step 1: Set up the equation
Since these angles form a linear pair:
\[ (3x - 17) + 122 = 180 \]

#### Step 2: Simplify the equation
\[ 3x - 17 + 122 = 180 \]
\[ 3x + 105 = 180 \]

#### Step 3: Solve for \(x\)
\[ 3x = 180 - 105 \]
\[ 3x = 75 \]
\[ x = \frac{75}{3} \]
\[ x = 25 \]

#### Answer:
\[ x = 25 \]

---

Problem 6:

\[ 80^\circ \quad \text{and} \quad 8x - 36 \]

#### Step 1: Set up the equation
Since these angles form a linear pair:
\[ 80 + (8x - 36) = 180 \]

#### Step 2: Simplify the equation
\[ 80 + 8x - 36 = 180 \]
\[ 8x + 44 = 180 \]

#### Step 3: Solve for \(x\)
\[ 8x = 180 - 44 \]
\[ 8x = 136 \]
\[ x = \frac{136}{8} \]
\[ x = 17 \]

#### Answer:
\[ x = 17 \]

---

Problem 7:

\[ 4x - 30 \quad \text{and} \quad 106^\circ \]

#### Step 1: Set up the equation
Since these angles form a linear pair:
\[ (4x - 30) + 106 = 180 \]

#### Step 2: Simplify the equation
\[ 4x - 30 + 106 = 180 \]
\[ 4x + 76 = 180 \]

#### Step 3: Solve for \(x\)
\[ 4x = 180 - 76 \]
\[ 4x = 104 \]
\[ x = \frac{104}{4} \]
\[ x = 26 \]

#### Answer:
\[ x = 26 \]

---

Problem 8:

\[ x + 40 \quad \text{and} \quad 52^\circ \]

#### Step 1: Set up the equation
Since these angles form a linear pair:
\[ (x + 40) + 52 = 180 \]

#### Step 2: Simplify the equation
\[ x + 40 + 52 = 180 \]
\[ x + 92 = 180 \]

#### Step 3: Solve for \(x\)
\[ x = 180 - 92 \]
\[ x = 88 \]

#### Answer:
\[ x = 88 \]

---

Final Answers:

\[
\boxed{
\begin{array}{cccc}
x = 55 & x = 8 & x = 22 & x = 16 \\
x = 25 & x = 17 & x = 26 & x = 88 \\
\end{array}
}
\]