Question

∠sut and ∠wxz are supplementary. complete the proof that \overleftrightarrow{tv} \parallel \overleftrightarrow{wy}.

statement	reason
2 m∠sut + m∠wxz = 180°	angles forming a linear pair sum to 180°; definition of supplementary angles
3 m∠sut + m∠tux = 180°
4 m∠sut + m∠wxz = m∠sut + m∠tux
5 m∠wxz = m∠tux	properties of addition, subtraction, multiplication, and division

Explanation:

Step1: Analyze Statement 2

To find the reason for \( m\angle SUT + m\angle WXZ = 180^\circ \), we recall the definition of supplementary angles (given that \( \angle SUT \) and \( \angle WXZ \) are supplementary). Wait, no—actually, \( \angle SUT \) and \( \angle TUX \) form a linear pair? Wait, no, the diagram: \( \angle SUT \) and \( \angle WXZ \) are supplementary (given). Wait, no, the first statement is " \( \angle SUT \) and \( \angle WXZ \) are supplementary" (given). Then statement 2: \( m\angle SUT + m\angle WXZ = 180^\circ \) is by the **definition of supplementary angles** (supplementary angles sum to \( 180^\circ \)). Wait, but the options include "Angles forming a linear pair sum to \( 180^\circ \)", "Definition of supplementary angles", etc. Wait, let's check the next steps.

Step2: Analyze Statement 3

Statement 3: \( m\angle SUT + m\angle TUX = 180^\circ \). Why? Because \( \angle SUT \) and \( \angle TUX \) form a linear pair (they are adjacent and form a straight line), so their sum is \( 180^\circ \). So the reason for statement 3 is "Angles forming a linear pair sum to \( 180^\circ \)".

Step3: Analyze Statement 4

Statement 4: \( m\angle SUT + m\angle WXZ = m\angle SUT + m\angle TUX \). This is by the **Substitution Property** (since \( m\angle SUT + m\angle WXZ = 180^\circ \) and \( m\angle SUT + m\angle TUX = 180^\circ \), so we can set them equal). Wait, but the options include "Vertical Angle Theorem" (no, vertical angles are equal, not here) or "Properties of equality" (addition/subtraction properties). Wait, actually, from statement 2 (\( m\angle SUT + m\angle WXZ = 180^\circ \)) and statement 3 (\( m\angle SUT + m\angle TUX = 180^\circ \)), we can use the **Transitive Property** or substitution to get \( m\angle SUT + m\angle WXZ = m\angle SUT + m\angle TUX \), then subtract \( m\angle SUT \) (subtraction property of equality) to get \( m\angle WXZ = m\angle TUX \) (statement 5). Then, \( \angle WXZ \) and \( \angle TUX \) are corresponding angles? Wait, no—\( \angle TUX \) and \( \angle WXZ \) being equal would imply \( TV \parallel WY \) by corresponding angles.

Wait, let's re-express the proof steps:

1. \( \angle SUT \) and \( \angle WXZ \) are supplementary (Given).
2. \( m\angle SUT + m\angle WXZ = 180^\circ \) (**Definition of supplementary angles**).
3. \( \angle SUT \) and \( \angle TUX \) form a linear pair (so \( m\angle SUT + m\angle TUX = 180^\circ \)) (**Angles forming a linear pair sum to \( 180^\circ \)**).
4. \( m\angle SUT + m\angle WXZ = m\angle SUT + m\angle TUX \) (**Substitution Property** or **Transitive Property**—since both equal \( 180^\circ \)).
5. \( m\angle WXZ = m\angle TUX \) (**Subtraction Property of Equality**—subtract \( m\angle SUT \) from both sides).
6. \( \angle WXZ \cong \angle TUX \) (Definition of congruent angles, since their measures are equal).
7. \( TV \parallel WY \) (Corresponding Angles Converse—if corresponding angles are equal, lines are parallel).

Wait, but the table has:

- Statement 2: \( m\angle SUT + m\angle WXZ = 180^\circ \)
Reason: **Definition of supplementary angles** (since \( \angle SUT \) and \( \angle WXZ \) are supplementary, their measures sum to \( 180^\circ \)).

- Statement 3: \( m\angle SUT + m\angle TUX = 180^\circ \)
Reason: **Angles forming a linear pair sum to \( 180^\circ \)** (because \( \angle SUT \) and \( \angle TUX \) are adjacent and form a straight line, so linear pair).

- Statement 4: \( m\angle SUT + m\angle WXZ = m\angle SUT + m\angle TUX \)
Reason: **Transitive Property** (or Substitution, since both equal \( 180^\circ \))—but the options include "Vertical Angle Theorem" (no) or "Properties of equality" (addition/subtraction/multiplication/division). Wait, the "Properties of addition, subtraction, multiplication, and division" (i.e., equality properties) allow us to set equal things equal.

- Statement 5: \( m\angle WXZ = m\angle TUX \)
Reason: **Subtraction Property of Equality** (subtract \( m\angle SUT \) from both sides of \( m\angle SUT + m\angle WXZ = m\angle SUT + m\angle TUX \)).

Then, since \( \angle WXZ \) and \( \angle TUX \) are corresponding angles (if \( TV \) and \( WY \) are cut by transversal \( SZ \)), their equality implies \( TV \parallel WY \) by Corresponding Angles Converse.

But let's focus on the blanks. Let's assume we need to fill the reasons:

1. Reason: Given (correct).
2. Reason: Definition of supplementary angles (since supplementary angles sum to \( 180^\circ \)).
3. Reason: Angles forming a linear pair sum to \( 180^\circ \) (because \( \angle SUT \) and \( \angle TUX \) are a linear pair).
4. Reason: Substitution Property (or Transitive Property, since both sums equal \( 180^\circ \)).
5. Reason: Subtraction Property of Equality (subtract \( m\angle SUT \) from both sides).

Wait, but the problem is to complete the proof that \( \overleftrightarrow{TV} \parallel \overleftrightarrow{WY} \). Let's structure the steps properly:

Step 1: Statement 1

\( \angle SUT \) and \( \angle WXZ \) are supplementary.
Reason: Given.

Step 2: Statement 2

\( m\angle SUT + m\angle WXZ = 180^\circ \).
Reason: Definition of supplementary angles (supplementary angles have a sum of \( 180^\circ \)).

Step 3: Statement 3

\( m\angle SUT + m\angle TUX = 180^\circ \).
Reason: Angles forming a linear pair sum to \( 180^\circ \) ( \( \angle SUT \) and \( \angle TUX \) form a linear pair, so they are adjacent and form a straight line, hence their sum is \( 180^\circ \) ).

Step 4: Statement 4

\( m\angle SUT + m\angle WXZ = m\angle SUT + m\angle TUX \).
Reason: Transitive Property of Equality (since both \( m\angle SUT + m\angle WXZ \) and \( m\angle SUT + m\angle TUX \) equal \( 180^\circ \), they are equal to each other).

Step 5: Statement 5

\( m\angle WXZ = m\angle TUX \).
Reason: Subtraction Property of Equality (subtract \( m\angle SUT \) from both sides of \( m\angle SUT + m\angle WXZ = m\angle SUT + m\angle TUX \)).

Step 6: Statement 6

\( \angle WXZ \cong \angle TUX \).
Reason: Definition of congruent angles (angles with equal measures are congruent).

Step 7: Statement 7

\( \overleftrightarrow{TV} \parallel \overleftrightarrow{WY} \).
Reason: Corresponding Angles Converse (if two lines are cut by a transversal and the corresponding angles are congruent, then the lines are parallel; here, \( \angle WXZ \) and \( \angle TUX \) are corresponding angles).

Now, let's match with the given table:

- Statement 2: Reason is "Definition of supplementary angles".
- Statement 3: Reason is "Angles forming a linear pair sum to \( 180^\circ \)".
- Statement 4: Reason is "Transitive Property" (or substitution, part of properties of equality).
- Statement 5: Reason is "Subtraction Property of Equality".

But the problem is to complete the proof, so the key steps are using the linear pair, supplementary angles, and corresponding angles converse. The final conclusion is that \( \overleftrightarrow{TV} \parallel \overleftrightarrow{WY} \) by Corresponding Angles Converse, using the equality of \( \angle WXZ \) and \( \angle TUX \) (corresponding angles) from the previous steps.

Answer:

To complete the proof:

1. Statement 2 Reason: Definition of supplementary angles (since \( \angle SUT \) and \( \angle WXZ \) are supplementary, their measures sum to \( 180^\circ \)).
2. Statement 3 Reason: Angles forming a linear pair sum to \( 180^\circ \) ( \( \angle SUT \) and \( \angle TUX \) form a linear pair).
3. Statement 4 Reason: Transitive Property (or Substitution) (since both sums equal \( 180^\circ \)).
4. Statement 5 Reason: Subtraction Property of Equality (subtract \( m\angle SUT \) from both sides).
5. Final Statement Reason: Corresponding Angles Converse ( \( \angle WXZ \cong \angle TUX \) implies \( \overleftrightarrow{TV} \parallel \overleftrightarrow{WY} \) ).

Thus, the proof shows \( \overleftrightarrow{TV} \parallel \overleftrightarrow{WY} \) by establishing corresponding angles \( \angle WXZ \) and \( \angle TUX \) are congruent, using supplementary angles, linear pair, and properties of equality.

(Note: If the question is to fill the blanks in the table, the reasons are as above. The final answer is that \( \overleftrightarrow{TV} \parallel \overleftrightarrow{WY} \) by the Corresponding Angles Converse.)

\(\boldsymbol{\overleftrightarrow{TV} \parallel \overleftrightarrow{WY}}\) (proven by Corresponding Angles Converse, using \( m\angle WXZ = m\angle TUX \) from the steps).