Geometry worksheet focusing on properties of equality and congruence with exercises and a word bank.

Problem Description:

The task involves determining the property of equality expressed by each of the following statements and writing it in the blank. The properties of equality include:

1. Addition Property of Equality: If $ a = b $, then $ a + c = b + c $.
2. Subtraction Property of Equality: If $ a = b $, then $ a - c = b - c $.
3. Multiplication Property of Equality: If $ a = b $, then $ ac = bc $.
4. Division Property of Equality: If $ a = b $ and $ c \neq 0 $, then $ \frac{a}{c} = \frac{b}{c} $.
5. Reflexive Property of Equality: $ a = a $.
6. Symmetric Property of Equality: If $ a = b $, then $ b = a $.
7. Transitive Property of Equality: If $ a = b $ and $ b = c $, then $ a = c $.

We are also required to solve the problem and explain the solution step by step.

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Solution:

#### Part 1: Determine the Property of Equality

We will analyze each statement and identify the property of equality it represents.

1. If $ a = b $, then $ b = a $.
- This is the Symmetric Property of Equality.
- Answer: Symmetric Property of Equality (SPE).

2. If $ a = b $, then $ a + c = b + c $.
- This is the Addition Property of Equality.
- Answer: Addition Property of Equality (APE).

3. If $ a = b $, then $ a - c = b - c $.
- This is the Subtraction Property of Equality.
- Answer: Subtraction Property of Equality (SPE).

4. If $ a = b $, then $ ac = bc $.
- This is the Multiplication Property of Equality.
- Answer: Multiplication Property of Equality (MPE).

5. If $ a = b $ and $ c \neq 0 $, then $ \frac{a}{c} = \frac{b}{c} $.
- This is the Division Property of Equality.
- Answer: Division Property of Equality (DPE).

6. If $ a = b $ and $ b = c $, then $ a = c $.
- This is the Transitive Property of Equality.
- Answer: Transitive Property of Equality (TPE).

7. If $ a = a $, then $ a = a $.
- This is the Reflexive Property of Equality.
- Answer: Reflexive Property of Equality (RPE).

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#### Part 2: Solve the Equations Using Properties of Equality

We are given several equations to solve, and we need to identify the property used in each step.

1. Solve $ x + 3 = 7 $.
- Step 1: Subtract 3 from both sides.
$$
x + 3 - 3 = 7 - 3
$$
$$
x = 4
$$
- Property used: Subtraction Property of Equality.
- Answer: SPE.

2. Solve $ 2x = 10 $.
- Step 1: Divide both sides by 2.
$$
\frac{2x}{2} = \frac{10}{2}
$$
$$
x = 5
$$
- Property used: Division Property of Equality.
- Answer: DPE.

3. Solve $ x - 5 = 2 $.
- Step 1: Add 5 to both sides.
$$
x - 5 + 5 = 2 + 5
$$
$$
x = 7
$$
- Property used: Addition Property of Equality.
- Answer: APE.

4. Solve $ \frac{x}{4} = 3 $.
- Step 1: Multiply both sides by 4.
$$
4 \cdot \frac{x}{4} = 4 \cdot 3
$$
$$
x = 12
$$
- Property used: Multiplication Property of Equality.
- Answer: MPE.

5. Solve $ 3x + 2 = 11 $.
- Step 1: Subtract 2 from both sides.
$$
3x + 2 - 2 = 11 - 2
$$
$$
3x = 9
$$
- Step 2: Divide both sides by 3.
$$
\frac{3x}{3} = \frac{9}{3}
$$
$$
x = 3
$$
- Properties used: Subtraction Property of Equality and Division Property of Equality.
- Answer: SPE, DPE.

6. Solve $ 2(x - 1) = 8 $.
- Step 1: Divide both sides by 2.
$$
\frac{2(x - 1)}{2} = \frac{8}{2}
$$
$$
x - 1 = 4
$$
- Step 2: Add 1 to both sides.
$$
x - 1 + 1 = 4 + 1
$$
$$
x = 5
$$
- Properties used: Division Property of Equality and Addition Property of Equality.
- Answer: DPE, APE.

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Final Answer:

$$
\boxed{
\text{See detailed explanations above.}
}
$$