Question

wx || su, complete the proof that m∠tmx + m∠utw = 180°. image of a diagram with lines and angles, and a table with statements and reasons

Explanation:

Step 1: Analyze Statement 1

Statement 1 is \( \overleftrightarrow{WX} \parallel \overleftrightarrow{SU} \), which is given. So the reason for Statement 1 is "Given".

Step 2: Analyze Statement 2

Statement 2 is \( \angle TMX \cong \angle RTU \). Since \( \overleftrightarrow{WX} \parallel \overleftrightarrow{SU} \) and \( \overleftrightarrow{RY} \) is a transversal, these are corresponding angles. So the reason is "Corresponding Angles Theorem" (when lines are parallel, corresponding angles are congruent).

Step 3: Analyze Statement 3

Statement 3 is \( m\angle RTU + m\angle UTW = 180^\circ \). \( \angle RTU \) and \( \angle UTW \) form a linear pair (they are adjacent and form a straight line), so their sum is \( 180^\circ \). The reason is "Angles forming a linear pair sum to \( 180^\circ \)".

Step 4: Analyze Statement 4

Statement 4 is \( m\angle TMX + m\angle UTW = 180^\circ \). From Statement 2, \( \angle TMX \cong \angle RTU \), so by the "Substitution Property" (substituting \( \angle TMX \) for \( \angle RTU \) in Statement 3) or using the definition of congruent angles (\( m\angle TMX = m\angle RTU \)) and then substitution, we get this. Also, since we want to show they are supplementary, the reason can be "Substitution" (using \( m\angle TMX = m\angle RTU \) from congruence) or "Definition of Supplementary Angles" (since their sum is \( 180^\circ \)). But following the flow, after establishing \( \angle TMX \cong \angle RTU \) (so their measures are equal) and \( m\angle RTU + m\angle UTW = 180^\circ \), substituting gives \( m\angle TMX + m\angle UTW = 180^\circ \), so the reason is "Substitution" (or using the definition of supplementary angles as the final step).

To fill the table:

• Statement 1 Reason: Given
• Statement 2 Reason: Corresponding Angles Theorem
• Statement 3 Reason: Angles forming a linear pair sum to \( 180^\circ \)
• Statement 4 Reason: Substitution (or Definition of Supplementary Angles, since \( m\angle TMX + m\angle UTW = 180^\circ \) means they are supplementary)

If we follow the options:

For Statement 2: Corresponding Angles Theorem (matches the option "Corresponding Angles Theorem")

For Statement 3: Angles forming a linear pair sum to \( 180^\circ \) (matches that option)

For Statement 4: Since \( m\angle TMX = m\angle RTU \) (from congruence) and \( m\angle RTU + m\angle UTW = 180^\circ \), substituting gives \( m\angle TMX + m\angle UTW = 180^\circ \), so the reason is "Substitution" (or using the fact that if two angles are congruent to a third angle, or here substituting the equal measure, leading to supplementary angles).

But to complete the proof:

1. \( \overleftrightarrow{WX} \parallel \overleftrightarrow{SU} \) - Given
2. \( \angle TMX \cong \angle RTU \) - Corresponding Angles Theorem (parallel lines, transversal)
3. \( m\angle RTU + m\angle UTW = 180^\circ \) - Angles forming a linear pair sum to \( 180^\circ \)
4. \( m\angle TMX + m\angle UTW = 180^\circ \) - Substitution (since \( m\angle TMX = m\angle RTU \) from \( \angle TMX \cong \angle RTU \))

Answer:

To complete the proof, the reasons are:

1. Reason for Statement 1: Given
2. Reason for Statement 2: Corresponding Angles Theorem
3. Reason for Statement 3: Angles forming a linear pair sum to \( 180^\circ \)
4. Reason for Statement 4: Substitution (or Definition of Supplementary Angles, as \( m\angle TMX + m\angle UTW = 180^\circ \) shows they are supplementary)

(If selecting from the provided options, match each statement to the corresponding reason as above.)