Question

prove: ∠3 ≅ ∠7

statements

1. k || l
2. ∠3 and ∠4 are supplementary
3. m∠3 + m∠4 = 180
4. ∠4 and ∠7 are a linear pair
5. ∠4 and ∠7 are supplementary
6. m∠4 + m∠7 = 180
7. m∠3 + m∠4 = m∠4 + m∠7
8. m∠3 = m∠7
9. ∠3 ≅ ∠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

all of the statements are correct. 3 of the reasons are incorrect.
type which row is incorrect
number of incorrect reason: 2
number of reason: 9
number of correct reason:
type the corrected reason (use lowercase)
corrected reason:
corrected reason:
corrected reason:

Explanation:

To determine the incorrect row, we analyze each statement - reason pair:

Step 1: Analyze Row 1

• Statement: \( k \parallel l \)
• Reason: Parallel Lines Postulate.

This is valid (defines the parallel lines).

Step 2: Analyze Row 2

• Statement: \( \angle 3 \) and \( \angle 4 \) are supplementary.
• Reason: Same Side Interior Angles Postulate.

When lines are parallel, same - side interior angles are supplementary. Valid.

Step 3: Analyze Row 3

• Statement: \( m\angle 3 + m\angle 4 = 180^\circ \)
• Reason: Definition of Supplementary Angles.

Supplementary angles sum to \( 180^\circ \). Valid.

Step 4: Analyze Row 4

• Statement: \( \angle 4 \) and \( \angle 7 \) are a linear pair.
• Reason: Definition of Linear Pair.

A linear pair of angles are adjacent and form a straight line. If \( k\parallel l \), \( \angle 4 \) and \( \angle 7 \) are not adjacent (they are alternate - interior or corresponding? Wait, no, for a linear pair, they must share a common side and vertex and form a straight line. \( \angle 4 \) and \( \angle 7 \) are not a linear pair. But wait, let's check the reason. Wait, maybe the error is in row 5, 6, 7, 8, 9. Wait, let's re - check row 5:

Step 5: Analyze Row 5

• Statement: \( \angle 4 \) and \( \angle 7 \) are supplementary.
• Reason: Linear Pair Theorem.

The Linear Pair Theorem states that a linear pair of angles is supplementary. But \( \angle 4 \) and \( \angle 7 \) are not a linear pair (from row 4's incorrect statement). But actually, the error is in row 6:

Step 6: Analyze Row 6

• Statement: \( m\angle 4 + m\angle 7 = 180^\circ \)
• Reason: Definition of Complementary Angles.

Complementary angles sum to \( 90^\circ \), supplementary to \( 180^\circ \). This is incorrect. The reason should be related to supplementary angles, not complementary.

Step 7: Analyze Row 7

• Statement: \( m\angle 3 + m\angle 4 = m\angle 4 + m\angle 7 \)
• Reason: Addition Property.

If we subtract \( m\angle 4 \) from both sides, we get \( m\angle 3=m\angle 7 \), which is related to the Subtraction Property (Subtraction Property of Equality: if \( a + b=a + c \), then \( b = c \)). But the reason here is Addition Property, which is misapplied. Wait, no, let's check row 8:

Step 8: Analyze Row 8

• Statement: \( m\angle 3 = m\angle 7 \)
• Reason: Subtraction Property.

From \( m\angle 3 + m\angle 4=m\angle 4 + m\angle 7 \), subtracting \( m\angle 4 \) from both sides (Subtraction Property of Equality) gives \( m\angle 3 = m\angle 7 \). Valid.

Step 9: Analyze Row 9

• Statement: \( \angle 3\cong\angle 7 \)
• Reason: Definition of Congruent Angles.

Congruent angles have equal measures, so if \( m\angle 3 = m\angle 7 \), then \( \angle 3\cong\angle 7 \). Valid.

But the most obvious error is in row 6: the reason for \( m\angle 4 + m\angle 7 = 180^\circ \) is given as "Definition of Complementary Angles", but complementary angles sum to \( 90^\circ \), while supplementary angles sum to \( 180^\circ \). So the incorrect row is row 6.

Answer:

Row 6 (Statement: \( m\angle 4 + m\angle 7 = 180^\circ \), Reason: Definition of Complementary Angles) is incorrect. The correct reason should be related to supplementary angles (e.g., if \( \angle 4 \) and \( \angle 7 \) were supplementary, but actually, from the parallel lines, \( \angle 3\cong\angle 7 \) (alternate - interior angles), and \( \angle 3 \) and \( \angle 4 \) are supplementary, so \( \angle 4 \) and \( \angle 7 \) are supplementary. But the reason given in row 6 is wrong as it refers to complementary angles.