Segment and Angle Addition Postulate Notes and Worksheets ... - Free Printable

Let's solve each of these angle addition postulate problems step by step, filling in the reasons for each proof. We’ll use geometric reasoning and standard postulates.

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Problem 1

Given: $ m\angle 1 = m\angle 3 $
Prove: $ m\angle EFH = m\angle IFG $

#### Diagram:
- Point F is the vertex.
- Rays: FE, FI, FH, FG
- Angles: $ \angle 1 = \angle EFI $, $ \angle 2 = \angle IFH $, $ \angle 3 = \angle HFG $
- So: $ \angle EFH = \angle 1 + \angle 2 $, $ \angle IFG = \angle 2 + \angle 3 $

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| 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 |
| 4. $ m\angle EFH = m\angle IFG $ | 4. Substitution Property of Equality (from Step 2 and 3) |

✔ Explanation: Since $ \angle 1 = \angle 3 $, adding $ \angle 2 $ to both gives equal sums. By angle addition, those sums are the measures of $ \angle EFH $ and $ \angle IFG $, so they are equal.

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Problem 2

Given: $ m\angle PQR = 130^\circ $, $ m\angle PQS = 5x $, $ m\angle SQR = 30^\circ $
Prove: $ x = 20 $

#### Diagram:
- Ray QS splits $ \angle PQR $ into $ \angle PQS $ and $ \angle SQR $
- So $ \angle PQR = \angle PQS + \angle SQR $

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| Statements | Reasons |
|-----------|--------|
| 1. $ m\angle PQR = 130^\circ $, $ m\angle PQS = 5x $, $ m\angle SQR = 30^\circ $ | 1. Given |
| 2. $ m\angle PQR = m\angle PQS + m\angle SQR $ | 2. Angle Addition Postulate |
| 3. $ 130 = 5x + 30 $ | 3. Substitution (plug in values from Step 1) |
| 4. $ 130 - 30 = 5x + 30 - 30 $ | 4. Subtraction Property of Equality |
| 5. $ 100 = 5x $ | 5. Simplify |
| 6. $ \frac{100}{5} = \frac{5x}{5} $ | 6. Division Property of Equality |
| 7. $ x = 20 $ | 7. Simplify |

✔ Explanation: The total angle is split into two parts. Plug in expressions and solve for $ x $. The algebra follows basic properties.

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Problem 3

Given: $ m\angle GKI = m\angle HKJ $
Prove: $ m\angle 3 = m\angle 5 $

#### Diagram:
- Point K is vertex with rays KG, KH, KI, KJ
- Angles:
- $ \angle 3 = \angle GK H $
- $ \angle 4 = \angle HKI $
- $ \angle 5 = \angle IKJ $
- So: $ \angle GKI = \angle 3 + \angle 4 $, $ \angle HKJ = \angle 4 + \angle 5 $

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| 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 |
| 3. $ m\angle 4 + m\angle 5 = m\angle HKJ $ | 3. Angle Addition Postulate |
| 4. $ m\angle 3 + m\angle 4 = m\angle HKJ $ | 4. Substitution (Step 1 & 3) |
| 5. $ m\angle 3 + m\angle 4 = m\angle 4 + m\angle 5 $ | 5. Substitution (Steps 2 and 4) |
| 6. $ m\angle 3 = m\angle 5 $ | 6. Subtraction Property of Equality (subtract $ m\angle 4 $ from both sides) |

✔ Explanation: Both angles $ \angle GKI $ and $ \angle HKJ $ are equal and composed of $ \angle 4 $ plus another angle. Subtracting $ \angle 4 $ from both shows $ \angle 3 = \angle 5 $.

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Problem 4

Given: $ m\angle JKL = m\angle MNO $, $ m\angle 1 = m\angle 3 $
Prove: $ m\angle 2 = m\angle 4 $

#### Diagram:
- Two separate angles at points K and N
- $ \angle JKL = \angle 1 + \angle 2 $
- $ \angle MNO = \angle 3 + \angle 4 $
- Given that $ \angle JKL = \angle MNO $ and $ \angle 1 = \angle 3 $

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| 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 |
| 3. $ m\angle 1 + m\angle 2 = m\angle MNO $ | 3. Substitution (from Step 1 and 2) |
| 4. $ m\angle 1 + m\angle 2 = m\angle 3 + m\angle 4 $ | 4. Substitution (from Step 2 and 3) |
| 5. $ m\angle 1 + m\angle 2 = m\angle 1 + m\angle 4 $ | 5. Substitution (since $ m\angle 3 = m\angle 1 $) |
| 6. $ m\angle 2 = m\angle 4 $ | 6. Subtraction Property of Equality (subtract $ m\angle 1 $ from both sides) |

✔ Explanation: Since the total angles are equal and one part ($ \angle 1 $ and $ \angle 3 $) are equal, the other parts ($ \angle 2 $ and $ \angle 4 $) must also be equal.

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✔ Final Answer Summary:

All proofs rely on:
- Angle Addition Postulate: If a point lies in the interior of an angle, then the sum of the measures of the two adjacent angles equals the measure of the whole angle.
- Properties of Equality: Substitution, Addition, Subtraction, Division, etc.
- Given information as starting point.

Each problem uses logical steps to reach the conclusion using geometry postulates and algebraic reasoning.

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