Question

statements

1. ( k parallel ell )
2. ( angle 3 ) and ( angle 4 ) are supplementary
3. ( mangle 3 + mangle 4 = 180 )
4. ( angle 4 ) and ( angle 7 ) are a linear pair
5. ( angle 4 ) and ( angle 7 ) are supplementary
6. ( mangle 4 + mangle 7 = 180 )
7. ( mangle 3 + mangle 4 = mangle 4 + mangle 7 )
8. ( mangle 3 = mangle 7 )
9. ( angle 3 cong angle 7 )

reasons

1. parallel line postulate
2. same - side interior angles postulate
3. definition of supplementary angles
4. definition of linear pair
5. linear pair theorem
6. definition of complementary angles
7. addition property
8. subtraction property
9. definition of congruent angles

all of the statements are correct. 3 of the reasons are incorrect.

Explanation:

To identify the incorrect reasons, we analyze each step:

Step 1: Analyze Reason 1

• Statement: \( k \parallel \ell \)
• Reason: "Parallel Line Postulate" – This is vague. Typically, we state the given (e.g., "Given") or a specific postulate, but if \( k \parallel \ell \) is given, "Given" is better. However, let’s check other reasons first.

Step 2: Analyze Reason 6

• Statement: \( m\angle 4 + m\angle 7 = 180 \)
• Reason: "Definition of Complementary Angles" – Complementary angles sum to \( 90^\circ \), while supplementary angles sum to \( 180^\circ \). This is incorrect. It should be "Definition of Supplementary Angles."

Step 3: Analyze Reason 7

• Statement: \( m\angle 3 + m\angle 4 = m\angle 4 + m\angle 7 \)
• Reason: "Addition Property" – This is incorrect. The correct reason is "Substitution Property" (since \( m\angle 3 + m\angle 4 = 180 \) and \( m\angle 4 + m\angle 7 = 180 \), we substitute \( 180 \) with the other expression).

Step 4: Analyze Reason 8

• Statement: \( m\angle 3 = m\angle 7 \)
• Reason: "Subtraction Property" – To get \( m\angle 3 = m\angle 7 \) from \( m\angle 3 + m\angle 4 = m\angle 4 + m\angle 7 \), we subtract \( m\angle 4 \) from both sides. This is the Subtraction Property of Equality, so this reason is actually correct. Wait, recheck:

Wait, let’s re-express:
From \( m\angle 3 + m\angle 4 = m\angle 4 + m\angle 7 \), subtract \( m\angle 4 \) from both sides:
\( m\angle 3 + m\angle 4 - m\angle 4 = m\angle 4 + m\angle 7 - m\angle 4 \)
\( m\angle 3 = m\angle 7 \)
So "Subtraction Property" is correct.

Step 5: Recheck Reason 1

If \( k \parallel \ell \) is given, the reason should be "Given" (not "Parallel Line Postulate," which is too vague). So Reason 1 is incorrect.

Step 6: Recheck Reason 5

• Statement: \( \angle 4 \) and \( \angle 7 \) are supplementary
• Reason: "Linear Pair Theorem" – A linear pair of angles is supplementary (by definition), so "Linear Pair Theorem" (or "Linear Pair Postulate") is correct.

Final Incorrect Reasons:

1. Reason 1: Should be "Given" (not "Parallel Line Postulate").
2. Reason 6: Should be "Definition of Supplementary Angles" (not "Complementary").
3. Reason 7: Should be "Substitution Property" (not "Addition Property").

To confirm, the three incorrect reasons are:

• Reason 1 (Parallel Line Postulate)
• Reason 6 (Definition of Complementary Angles)
• Reason 7 (Addition Property)

Answer:

The three incorrect reasons are:

1. Reason 1 (Parallel Line Postulate)
2. Reason 6 (Definition of Complementary Angles)
3. Reason 7 (Addition Property)