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

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 properties like the Angle Addition Postulate, Substitution Property, Reflexive Property, Addition/Subtraction Properties, and Transitive Property.

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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. Transitive Property of Equality (from steps 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 $

#### 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 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 the equation and solve for $ x $ using algebraic properties.

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

> ✔ Explanation: Both large angles ($ \angle GKI $ and $ \angle HKJ $) are equal and composed of adjacent angles. Since $ \angle 4 $ is common, subtracting it from both sides leaves $ \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 $

#### 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 (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 $) |
| 6. $ m\angle 2 = m\angle 4 $ | 6. Subtraction Property of Equality (subtract $ m\angle 1 $ from both sides) |

> ✔ Explanation: Use angle addition to express both large angles in terms of smaller ones. Then substitute known equalities and simplify to show $ \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.
- Substitution Property: Replace equal quantities.
- Addition/Subtraction/Division Properties of Equality.
- Transitive Property: If $ a = b $ and $ b = c $, then $ a = c $.

Each proof logically builds from given information using valid geometric and algebraic reasoning.

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