Geometry worksheet practicing the Angle Addition Postulate with four proof problems.

Problem Analysis and Solution

The image contains four geometric proofs involving the Angle Addition Postulate. Each proof requires filling in the "Reasons" column for the given "Statements." Below, I will solve each problem step by step.

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Problem 1:

#### Given: \( m\angle 1 = m\angle 3 \)
#### Prove: \( m\angle EFH = m\angle IFG \)

| Statements | Reasons |
|------------|---------|
| 1. \( m\angle 1 = m\angle 3 \) | 1. Given |
| 2. \( m\angle 1 + m\angle 2 = m\angle 2 + m\angle 3 \) | 2. Addition Property of Equality (Add \( m\angle 2 \) to both sides) |
| 3. \( m\angle 1 + m\angle 2 = m\angle EFH \) and \( m\angle 2 + m\angle 3 = m\angle IFG \) | 3. Angle Addition Postulate (Sum of adjacent angles forms a larger angle) |
| 4. \( m\angle EFH = m\angle IFG \) | 4. Substitution Property of Equality (Substitute from statements 2 and 3) |

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Problem 2:

#### Given: \( m\angle PQR = 130^\circ \), \( m\angle PQS = 5x \), and \( m\angle SQR = 30^\circ \)
#### Prove: \( x = 20 \)

| Statements | Reasons |
|------------|---------|
| 1. \( m\angle PQR = 130^\circ \), \( m\angle PQS = 5x \), and \( m\angle SQR = 30^\circ \) | 1. Given |
| 2. \( m\angle PQR = m\angle PQS + m\angle SQR \) | 2. Angle Addition Postulate (Whole is the sum of its parts) |
| 3. \( 130 = 5x + 30 \) | 3. Substitution Property of Equality (Substitute the given values) |
| 4. \( 130 - 30 = 5x + 30 - 30 \) | 4. Subtraction Property of Equality (Subtract 30 from both sides) |
| 5. \( 100 = 5x \) | 5. Simplification |
| 6. \( \frac{100}{5} = \frac{5x}{5} \) | 6. Division Property of Equality (Divide both sides by 5) |
| 7. \( x = 20 \) | 7. Simplification |

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Problem 3:

#### Given: \( m\angle GKI = m\angle HKJ \)
#### Prove: \( m\angle 3 = m\angle 5 \)

| Statements | Reasons |
|------------|---------|
| 1. \( m\angle GKI = m\angle HKJ \) | 1. Given |
| 2. \( m\angle 3 + m\angle 4 = m\angle GKI \) | 2. Angle Addition Postulate (Sum of adjacent angles forms a larger angle) |
| 3. \( m\angle 4 + m\angle 5 = m\angle HKJ \) | 3. Angle Addition Postulate (Sum of adjacent angles forms a larger angle) |
| 4. \( m\angle 3 + m\angle 4 = m\angle HKJ \) | 4. Substitution Property of Equality (Substitute \( m\angle GKI = m\angle HKJ \)) |
| 5. \( m\angle 3 + m\angle 4 = m\angle 4 + m\angle 5 \) | 5. Transitive Property of Equality (From statements 3 and 4) |
| 6. \( m\angle 3 = m\angle 5 \) | 6. Subtraction Property of Equality (Subtract \( m\angle 4 \) from both sides) |

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Problem 4:

#### Given: \( m\angle JKL = m\angle MNO \) and \( m\angle 1 = m\angle 3 \)
#### Prove: \( m\angle 2 = m\angle 4 \)

| Statements | Reasons |
|------------|---------|
| 1. \( m\angle JKL = m\angle MNO \) and \( m\angle 1 = m\angle 3 \) | 1. Given |
| 2. \( m\angle 1 + m\angle 2 = m\angle JKL \) and \( m\angle 3 + m\angle 4 = m\angle MNO \) | 2. Angle Addition Postulate (Sum of adjacent angles forms a larger angle) |
| 3. \( m\angle 1 + m\angle 2 = m\angle MNO \) | 3. Substitution Property of Equality (Substitute \( m\angle JKL = m\angle MNO \)) |
| 4. \( m\angle 1 + m\angle 2 = m\angle 3 + m\angle 4 \) | 4. Substitution Property of Equality (Substitute \( m\angle 1 = m\angle 3 \)) |
| 5. \( m\angle 1 + m\angle 2 = m\angle 1 + m\angle 4 \) | 5. Transitive Property of Equality (From statements 4 and the given \( m\angle 1 = m\angle 3 \)) |
| 6. \( m\angle 2 = m\angle 4 \) | 6. Subtraction Property of Equality (Subtract \( m\angle 1 \) from both sides) |

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Final Answer:

\[
\boxed{
\begin{array}{c}
\text{Problem 1: } m\angle EFH = m\angle IFG \\
\text{Problem 2: } x = 20 \\
\text{Problem 3: } m\angle 3 = m\angle 5 \\
\text{Problem 4: } m\angle 2 = m\angle 4
\end{array}
}
\]