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 $ \angle EFH $ and $ \angle IFG $, so they must be 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 $

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| 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 |
| 3. $ 130 = 5x + 30 $ | 3. Substitution (plug in values from step 1 into step 2) |
| 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 $ \angle PQR $ is made up of $ \angle PQS $ and $ \angle SQR $. Set up an equation and solve algebraically.

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Problem 3

Given: $ m\angle GKI = m\angle HKJ $
Prove: $ m\angle 3 = m\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 (from step 1 and 3) |
| 5. $ m\angle 3 + m\angle 4 = m\angle 4 + m\angle 5 $ | 5. Substitution (step 4 and step 3) |
| 6. $ m\angle 3 = m\angle 5 $ | 6. Subtraction Property of Equality (subtract $ m\angle 4 $ from both sides) |

✔ Explanation: Since $ \angle GKI $ and $ \angle HKJ $ are equal, and both are sums involving $ \angle 4 $, we can subtract $ \angle 4 $ from both to show $ \angle 3 = \angle 5 $.

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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 $

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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: $ m\angle JKL = m\angle MNO $) |
| 4. $ m\angle 1 + m\angle 2 = m\angle 3 + m\angle 4 $ | 4. Substitution (use 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 $ from given) |
| 6. $ m\angle 2 = m\angle 4 $ | 6. Subtraction Property of Equality (subtract $ m\angle 1 $ from both sides) |

✔ Explanation: Use the given angle equality and angle addition to set up equations. Substitute known equalities and simplify to find $ \angle 2 = \angle 4 $.

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✔ Final Summary:

All proofs rely on:
- Angle Addition Postulate: The measure of an angle formed by two adjacent angles is the sum of their measures.
- Properties of Equality: Substitution, Addition, Subtraction, Division.
- Given information as starting points.

Each problem uses logical steps based on these principles.

Let me know if you'd like a visual explanation or any part clarified!