Preview of the Reasoning and Proof stations activity, featuring tasks on conditional statements, algebraic properties, and geometric proofs.

The image shows a "Stations Activity" focused on reasoning and proof in Geometry. The activity includes several stations, each with a specific task related to conditional statements, properties from algebra, and geometric proofs. Below, I will explain how to solve the tasks shown in the image.

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Station 3: Conditional Statements

#### Task:
Evaluate the converse, inverse, and contrapositive of the following conditional statement. Determine whether each is true or false. If false, provide a counterexample.

Conditional Statement:
"If you play a guitar, then you play a stringed instrument."

#### Solution:
1. Original Statement:
- If \( P \), then \( Q \)
\( P \): You play a guitar.
\( Q \): You play a stringed instrument.

2. Converse:
- If \( Q \), then \( P \)
"If you play a stringed instrument, then you play a guitar."
- Truth Value: False
Counterexample: Playing a violin (a stringed instrument) does not mean you play a guitar.

3. Inverse:
- If not \( P \), then not \( Q \)
"If you do not play a guitar, then you do not play a stringed instrument."
- Truth Value: False
Counterexample: Playing a violin (a stringed instrument) means you do not play a guitar.

4. Contrapositive:
- If not \( Q \), then not \( P \)
"If you do not play a stringed instrument, then you do not play a guitar."
- Truth Value: True
This is logically equivalent to the original statement.

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Station 5: Properties from Algebra

#### Task:
Fill in the reason for each step of solving the following equation:

\[ 4x - 7 = 3(x - 2) \]

#### Solution:
1. Step 1:
\[ 4x - 7 = 3(x - 2) \]
Reason: Given equation.

2. Step 2:
\[ 4x - 7 = 3x - 6 \]
Reason: Distributive Property (distribute 3 across \( x - 2 \)).

3. Step 3:
\[ 4x - 3x - 7 = -6 \]
Reason: Subtract \( 3x \) from both sides (Subtraction Property of Equality).

4. Step 4:
\[ x - 7 = -6 \]
Reason: Simplify (Combine like terms).

5. Step 5:
\[ x = 1 \]
Reason: Add 7 to both sides (Addition Property of Equality).

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Station 8: Proof

#### Task:
Complete the proof.

Given: \( OD \) is an angle bisector of \( \angle ABC \).
Prove: \( m\angle AOD = \frac{1}{2} m\angle ABC \)

#### Solution:
We are given that \( OD \) is the angle bisector of \( \angle ABC \). By definition, an angle bisector divides an angle into two congruent angles. Therefore, we can write the proof as follows:

| Statements | Reasons |
|-----------------|--------------|
| 1. \( OD \) is an angle bisector of \( \angle ABC \) | 1. Given |
| 2. \( \angle AOD \cong \angle COD \) | 2. Definition of angle bisector |
| 3. \( m\angle AOD = m\angle COD \) | 3. Congruent angles have equal measures |
| 4. \( m\angle ABC = m\angle AOD + m\angle COD \) | 4. Angle Addition Postulate |
| 5. \( m\angle ABC = m\angle AOD + m\angle AOD \) | 5. Substitution (from Step 3) |
| 6. \( m\angle ABC = 2m\angle AOD \) | 6. Combine like terms |
| 7. \( m\angle AOD = \frac{1}{2} m\angle ABC \) | 7. Division Property of Equality |

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Final Answer:

\[
\boxed{
\text{Station 3: Converse (False), Inverse (False), Contrapositive (True)} \\
\text{Station 5: Reasons are Given, Distributive Property, Subtraction Property of Equality, Simplify, Addition Property of Equality} \\
\text{Station 8: Proof steps as shown above}
}
\]