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 statement in the proofs. We'll use geometric reasoning, properties of equality, and the Angle Addition Postulate, which states:

> If point $ B $ lies in the interior of $ \angle AOC $, then
> $ m\angle AOB + m\angle BOC = m\angle AOC $.

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Problem 1

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

#### Proof Table:

| 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 steps 2 and 3) |

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Problem 2

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

#### Proof Table:

| 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 (replace angle measures with expressions) |
| 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 |

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Problem 3

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

#### Proof Table:

| 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) |
| 5. $ m\angle 3 + m\angle 4 = m\angle 4 + m\angle 5 $ | 5. Transitive Property of Equality (steps 3 & 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 $

#### Proof Table:

| 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 (from step 2 and 3) |
| 5. $ m\angle 1 + m\angle 2 = m\angle 1 + m\angle 4 $ | 5. Substitution (from step 1: $ 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) |

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✔ Summary of Key Concepts Used:

- Angle Addition Postulate: When two angles are adjacent, their sum equals the larger angle.
- Substitution Property: If $ a = b $, then $ a $ can be replaced with $ b $.
- Addition/Subtraction/Division Properties of Equality: You can perform operations on both sides of an equation.
- Transitive Property: If $ a = b $ and $ b = c $, then $ a = c $.

These proofs reinforce how algebraic reasoning is applied in geometry using angle relationships.

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