Algebra worksheet for solving variable values using balancing equations with geometric angle problems.

Problem Overview:

The task involves solving for the variables \( a \) and \( b \) in each of the given diagrams. Each diagram shows an angle labeled as \( XQR \), which is stated to be equal to \( 180^\circ \). The angles around point \( Q \) are expressed in terms of \( a \) and \( b \). We need to use the property that the sum of angles around a point is \( 360^\circ \) or, equivalently, that the angles on a straight line sum to \( 180^\circ \).

General Approach:

1. Identify the angles around point \( Q \): The angles around point \( Q \) are given in terms of \( a \) and \( b \).
2. Set up the equation: Since \( \angle XQR = 180^\circ \), the sum of the angles on either side of \( \angle XQR \) must equal \( 180^\circ \).
3. Solve the equation: Solve the resulting equation for the variables \( a \) and \( b \).

Solutions for Each Diagram:

#### 1)
- Angles around \( Q \): \( -3a + 56 \) and \( 5a + 40 \)
- Equation: \( -3a + 56 + 5a + 40 = 180 \)
- Simplify: \( 2a + 96 = 180 \)
- Solve for \( a \): \( 2a = 84 \implies a = 42 \)
- There is no \( b \) in this equation, so \( b \) is not defined here.
- Answer: \( a = 42 \), \( b = \text{undefined} \)

#### 2)
- Angles around \( Q \): \( 9b + 40 \) and \( 2b + 63 \)
- Equation: \( 9b + 40 + 2b + 63 = 180 \)
- Simplify: \( 11b + 103 = 180 \)
- Solve for \( b \): \( 11b = 77 \implies b = 7 \)
- There is no \( a \) in this equation, so \( a \) is not defined here.
- Answer: \( a = \text{undefined} \), \( b = 7 \)

#### 3)
- Angles around \( Q \): \( 35 + 2b \) and \( 2a + 25 \)
- Equation: \( 35 + 2b + 2a + 25 = 180 \)
- Simplify: \( 2a + 2b + 60 = 180 \)
- Solve for \( a \) and \( b \): \( 2a + 2b = 120 \implies a + b = 60 \)
- Without additional information, we cannot uniquely determine \( a \) and \( b \). However, if we assume the problem intends for us to solve for one variable in terms of the other, we can express \( b \) as \( b = 60 - a \).
- Answer: \( a + b = 60 \)

#### 4)
- Angles around \( Q \): \( 17 + 4b \) and \( 5b + 105 \)
- Equation: \( 17 + 4b + 5b + 105 = 180 \)
- Simplify: \( 9b + 122 = 180 \)
- Solve for \( b \): \( 9b = 58 \implies b = \frac{58}{9} \)
- There is no \( a \) in this equation, so \( a \) is not defined here.
- Answer: \( a = \text{undefined} \), \( b = \frac{58}{9} \)

#### 5)
- Angles around \( Q \): \( 6a + 79 \) and \( 3a + 43 \)
- Equation: \( 6a + 79 + 3a + 43 = 180 \)
- Simplify: \( 9a + 122 = 180 \)
- Solve for \( a \): \( 9a = 58 \implies a = \frac{58}{9} \)
- There is no \( b \) in this equation, so \( b \) is not defined here.
- Answer: \( a = \frac{58}{9} \), \( b = \text{undefined} \)

#### 6)
- Angles around \( Q \): \( -1b + 46 \) and \( 6 + 5b \)
- Equation: \( -b + 46 + 6 + 5b = 180 \)
- Simplify: \( 4b + 52 = 180 \)
- Solve for \( b \): \( 4b = 128 \implies b = 32 \)
- There is no \( a \) in this equation, so \( a \) is not defined here.
- Answer: \( a = \text{undefined} \), \( b = 32 \)

#### 7)
- Angles around \( Q \): \( 3a + 12 \) and \( 2a + 8 \)
- Equation: \( 3a + 12 + 2a + 8 = 180 \)
- Simplify: \( 5a + 20 = 180 \)
- Solve for \( a \): \( 5a = 160 \implies a = 32 \)
- There is no \( b \) in this equation, so \( b \) is not defined here.
- Answer: \( a = 32 \), \( b = \text{undefined} \)

#### 8)
- Angles around \( Q \): \( -9a + 139 \) and \( -3b + 59 \)
- Equation: \( -9a + 139 + (-3b + 59) = 180 \)
- Simplify: \( -9a - 3b + 198 = 180 \)
- Solve for \( a \) and \( b \): \( -9a - 3b = -18 \implies 3a + b = 6 \)
- Without additional information, we cannot uniquely determine \( a \) and \( b \). However, if we assume the problem intends for us to solve for one variable in terms of the other, we can express \( b \) as \( b = 6 - 3a \).
- Answer: \( 3a + b = 6 \)

Final Answers:

1. \( a = 42 \), \( b = \text{undefined} \)
2. \( a = \text{undefined} \), \( b = 7 \)
3. \( a + b = 60 \)
4. \( a = \text{undefined} \), \( b = \frac{58}{9} \)
5. \( a = \frac{58}{9} \), \( b = \text{undefined} \)
6. \( a = \text{undefined} \), \( b = 32 \)
7. \( a = 32 \), \( b = \text{undefined} \)
8. \( 3a + b = 6 \)

\[
\boxed{
\begin{array}{ll}
1. & a = 42, b = \text{undefined} \\
2. & a = \text{undefined}, b = 7 \\
3. & a + b = 60 \\
4. & a = \text{undefined}, b = \frac{58}{9} \\
5. & a = \frac{58}{9}, b = \text{undefined} \\
6. & a = \text{undefined}, b = 32 \\
7. & a = 32, b = \text{undefined} \\
8. & 3a + b = 6 \\
\end{array}
}
\]