Question

rewrite the two - column proof without using the congruent supplements theorem.

statements

1. blank
2. ( angle5 ) and ( angle6 ) are a linear pair. ( angle6 ) and ( angle7 ) are a linear pair.
3. ( angle5 ) and ( angle6 ) are supplementary. ( angle6 ) and ( angle7 ) are supplementary.
4. blank and blank
5. ( mangle5 + mangle6 = mangle6 + mangle7 )
6. blank

reasons

1. given
2. blank
3. blank
4. definition of supplementary angles
5. blank
6. subtraction property of equality

Explanation:

Step1: Identify Given (Missing in Statement 1)

Assume the given is about angles related to ∠5, ∠6, ∠7 (e.g., ∠5 and ∠7 related, but likely the initial given is context for linear pairs. Let's fill based on typical proof:

1. Statement 1: Let's say "∠5 and ∠7 are such that ∠5, ∠6 linear pair; ∠6, ∠7 linear pair" (but actually, the first given is usually the setup, like "∠5 and ∠6 form a linear pair, ∠6 and ∠7 form a linear pair" but no—wait, the problem is to rewrite the proof. Let's structure each step:

Step2: Reason for Linear Pair (Statement 2)

1. Reason: Definition of linear pair (adjacent angles forming a straight line).

Step3: Reason for Supplementary (Statement 3)

1. Reason: Linear Pair Postulate (linear pairs are supplementary).

Step4: Statement from Supplementary Def (Statement 4)

1. Statement: \( m\angle5 + m\angle6 = 180^\circ \) and \( m\angle6 + m\angle7 = 180^\circ \) (by definition of supplementary: sum to \( 180^\circ \)).

Step5: Reason for Equation (Statement 5)

1. Reason: Substitution Property (since both \( m\angle5 + m\angle6 \) and \( m\angle6 + m\angle7 \) equal \( 180^\circ \), they equal each other).

Step6: Statement from Subtraction (Statement 6)

1. Statement: \( m\angle5 = m\angle7 \) (subtract \( m\angle6 \) from both sides: \( m\angle5 + m\angle6 - m\angle6 = m\angle6 + m\angle7 - m\angle6 \)).

Filled Proof Table:

STATEMENTS	REASONS
2. ∠5 and ∠6 are a linear pair. ∠6 and ∠7 are a linear pair.	2. Definition of Linear Pair
3. ∠5 and ∠6 are supplementary. ∠6 and ∠7 are supplementary.	3. Linear Pair Postulate
4. \( m\angle5 + m\angle6 = 180^\circ \) and \( m\angle6 + m\angle7 = 180^\circ \)	4. Definition of Supplementary Angles
5. \( m\angle5 + m\angle6 = m\angle6 + m\angle7 \)	5. Substitution Property (both equal \( 180^\circ \))
6. \( m\angle5 = m\angle7 \)	6. Subtraction Property of Equality

(Note: The exact "given" in Statement 1 depends on the original problem’s context, but this follows the logical flow of proving \( \angle5 \cong \angle7 \) via linear pairs and equality properties, avoiding Congruent Supplements Theorem.)

Answer:

Step1: Identify Given (Missing in Statement 1)

Assume the given is about angles related to ∠5, ∠6, ∠7 (e.g., ∠5 and ∠7 related, but likely the initial given is context for linear pairs. Let's fill based on typical proof:

1. Statement 1: Let's say "∠5 and ∠7 are such that ∠5, ∠6 linear pair; ∠6, ∠7 linear pair" (but actually, the first given is usually the setup, like "∠5 and ∠6 form a linear pair, ∠6 and ∠7 form a linear pair" but no—wait, the problem is to rewrite the proof. Let's structure each step:

Step2: Reason for Linear Pair (Statement 2)

1. Reason: Definition of linear pair (adjacent angles forming a straight line).

Step3: Reason for Supplementary (Statement 3)

1. Reason: Linear Pair Postulate (linear pairs are supplementary).

Step4: Statement from Supplementary Def (Statement 4)

1. Statement: \( m\angle5 + m\angle6 = 180^\circ \) and \( m\angle6 + m\angle7 = 180^\circ \) (by definition of supplementary: sum to \( 180^\circ \)).

Step5: Reason for Equation (Statement 5)

1. Reason: Substitution Property (since both \( m\angle5 + m\angle6 \) and \( m\angle6 + m\angle7 \) equal \( 180^\circ \), they equal each other).

Step6: Statement from Subtraction (Statement 6)

1. Statement: \( m\angle5 = m\angle7 \) (subtract \( m\angle6 \) from both sides: \( m\angle5 + m\angle6 - m\angle6 = m\angle6 + m\angle7 - m\angle6 \)).

Filled Proof Table:

STATEMENTS	REASONS
2. ∠5 and ∠6 are a linear pair. ∠6 and ∠7 are a linear pair.	2. Definition of Linear Pair
3. ∠5 and ∠6 are supplementary. ∠6 and ∠7 are supplementary.	3. Linear Pair Postulate
4. \( m\angle5 + m\angle6 = 180^\circ \) and \( m\angle6 + m\angle7 = 180^\circ \)	4. Definition of Supplementary Angles
5. \( m\angle5 + m\angle6 = m\angle6 + m\angle7 \)	5. Substitution Property (both equal \( 180^\circ \))
6. \( m\angle5 = m\angle7 \)	6. Subtraction Property of Equality

(Note: The exact "given" in Statement 1 depends on the original problem’s context, but this follows the logical flow of proving \( \angle5 \cong \angle7 \) via linear pairs and equality properties, avoiding Congruent Supplements Theorem.)