Question

type which row is incorrect
all of the statements are correct. 3 of the reasons are incorrect.
type the corrected reason (use words)
statements:

1. ( a parallel t ) (or ( ell parallel t ))
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 lines 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

Explanation:

Step1: Analyze each row's reason

- Row 6: Statement is \( m\angle4 + m\angle7 = 180 \), reason given is "Definition of Complementary Angles". But complementary angles sum to \( 90^\circ \), supplementary sum to \( 180^\circ \). So this reason is wrong.
- Row 5: Statement is \( \angle4 \) and \( \angle7 \) are supplementary, reason "Linear Pair Theorem". Linear Pair Theorem says linear pair angles are supplementary, but \( \angle4 \) and \( \angle7 \) (if \( a\parallel t \)): Wait, no, let's check row 6 first. Wait, row 6's reason is incorrect because complementary angles add to \( 90 \), not \( 180 \). So row 6 has incorrect reason. Also, row 5: If \( \angle4 \) and \( \angle7 \) are supplementary, but "Linear Pair Theorem" applies to adjacent angles forming a line. Wait, maybe row 6 is more obvious. Wait, the problem says "3 of the reasons are incorrect". Let's check each:
1. Row 1: \( a\parallel t \), reason "Parallel Lines Postulate" – correct.
2. Row 2: \( \angle3 \) and \( \angle4 \) supplementary, reason "Same - Side Interior Angles Postulate" (if \( a\parallel t \), same - side interior angles are supplementary) – correct.
3. Row 3: \( m\angle3 + m\angle4 = 180 \), reason "Definition of Supplementary Angles" – correct (supplementary angles sum to \( 180 \)).
4. Row 4: \( \angle4 \) and \( \angle7 \) are a linear pair, reason "Definition of Linear Pair" – correct (linear pair are adjacent, form a line).
5. Row 5: \( \angle4 \) and \( \angle7 \) are supplementary, reason "Linear Pair Theorem" – if they are a linear pair (from row 4), then Linear Pair Theorem says they are supplementary – correct? Wait, no, row 4 says they are a linear pair, row 5 says they are supplementary, reason Linear Pair Theorem – that's correct. Wait, row 6: \( m\angle4 + m\angle7 = 180 \), reason "Definition of Complementary Angles" – wrong, should be "Definition of Supplementary Angles" or "Linear Pair Theorem".
6. Row 7: \( m\angle3 + m\angle4 = m\angle4 + m\angle7 \), reason "Addition Property" – actually, it's Subtraction Property (subtract \( m\angle4 \) from both sides), so wrong.
7. Row 8: \( m\angle3 = m\angle7 \), reason "Subtraction Property" – from row 7, subtracting \( m\angle4 \) from both sides, so correct (Subtraction Property of Equality). Wait, no: row 7 is \( m\angle3 + m\angle4 = m\angle4 + m\angle7 \), subtract \( m\angle4 \) from both sides: \( m\angle3 = m\angle7 \), so reason for row 7 should be "Subtraction Property", and row 8's reason: if \( m\angle3 = m\angle7 \), then \( \angle3\cong\angle7 \) by Definition of Congruent Angles (row 9 is correct). Wait, row 7's reason is "Addition Property" – wrong, should be "Subtraction Property". Row 6's reason is wrong (complementary vs supplementary). Row 5: Wait, maybe I made a mistake. Let's re - evaluate:

Wait, the problem says "Type which row is incorrect" (maybe one row first? Wait, the image shows "Type which row is incorrect" and "Type the corrected reason". Let's look at row 6:

Statement: \( m\angle4 + m\angle7 = 180 \)

Reason: Definition of Complementary Angles

Complementary angles are two angles whose sum is \( 90^\circ \), supplementary angles sum to \( 180^\circ \). So the correct reason should be "Definition of Supplementary Angles" or "Linear Pair Theorem" (since \( \angle4 \) and \( \angle7 \) are a linear pair from row 4, and linear pair angles are supplementary). So row 6 has an incorrect reason.

Step2: Determine the incorrect row and corrected reason

The row with incorrect reason is row 6. The corrected reason should be "Definition of Supplementary Angles" (or "Linear Pair Theorem" since \( \angle4 \) and \( \angle7 \) are a linear pair and linear pair angles are supplementary).

Answer:

Incorrect row: 6
Corrected reason: Definition of Supplementary Angles (or Linear Pair Theorem)