Math worksheet for solving equations using the linear pair theorem.

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.

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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:
\[ x = 55 \]

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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:
\[ x = 16 \]

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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:
\[ x = 88 \]

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Final Answers:

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