Question

su || vx. complete the proof that ( mangle stw + mangle twv = 180^circ ).

statement	reason
2. ( angle twv cong angle rts )
3. ( mangle rts + mangle stw = 180^circ )
4. ( mangle stw + mangle twv = 180^circ )

reason options: angles forming a linear pair sum to ( 180^circ ), converse of corresponding angles theorem, corresponding angles theorem, definition of congruence, definition of supplementary angles, given.

diagram: lines ( ry ), ( su ), ( vx ) (with ( su parallel vx )) intersecting at points ( t ), ( w ); arrows indicate parallelism/direction.

Explanation:

To solve this proof, we analyze each statement and its corresponding reason:

Statement 1: \( SU \parallel VX \)

• Reason: Given (this is the initial information provided, so the reason is "Given").

Statement 2: \( \angle TWV \cong \angle RTS \)

• Reason: Corresponding Angles Theorem (when two parallel lines are cut by a transversal, corresponding angles are congruent. Here, \( SU \parallel VX \) and \( RY \) is the transversal, so \( \angle TWV \) and \( \angle RTS \) are corresponding angles).

Statement 3: \( m\angle RTS + m\angle STW = 180^\circ \)

• Reason: Angles forming a linear pair sum to \( 180^\circ \) ( \( \angle RTS \) and \( \angle STW \) form a linear pair, so their measures add up to \( 180^\circ \)).

Statement 4: \( m\angle STW + m\angle TWV = 180^\circ \)

• Reason: Substitution Property (we substitute \( \angle RTS \) with \( \angle TWV \) because they are congruent from Statement 2, so \( m\angle RTS + m\angle STW = 180^\circ \) becomes \( m\angle STW + m\angle TWV = 180^\circ \)).

For the blank in the "Reason" column corresponding to the first part (if we assume the first blank is for Statement 2, but looking at the dropdown, let's match:

Wait, the dropdown options are: Angles forming a linear pair sum to \( 180^\circ \), Converse of Corresponding Angles Theorem, Corresponding Angles Theorem, Definition of congruence, Definition of supplementary angles, Given.

Wait, let's re - evaluate:

1. \( SU \parallel VX \): Reason is "Given".
2. \( \angle TWV \cong \angle RTS \): Reason is "Corresponding Angles Theorem" (since \( SU \parallel VX \) and transversal \( RY \), corresponding angles are congruent).
3. \( m\angle RTS + m\angle STW = 180^\circ \): Reason is "Angles forming a linear pair sum to \( 180^\circ \)".
4. \( m\angle STW + m\angle TWV = 180^\circ \): Reason is "Substitution" (but since substitution is not in the list, wait, maybe the blank is for the first reason? Wait, the problem says "Complete the proof that \( m\angle STW + m\angle TWV = 180^\circ \)". Let's check the statements and reasons again.

Wait, the first statement is \( SU \parallel VX \) with reason "Given". The second statement is \( \angle TWV \cong \angle RTS \) with reason "Corresponding Angles Theorem" (because when two parallel lines are cut by a transversal, corresponding angles are congruent). The third statement is \( m\angle RTS + m\angle STW = 180^\circ \) with reason "Angles forming a linear pair sum to \( 180^\circ \)". The fourth statement is \( m\angle STW + m\angle TWV = 180^\circ \) with reason "Substitution" (since \( \angle TWV \cong \angle RTS \), we can substitute \( \angle RTS \) with \( \angle TWV \) in the equation \( m\angle RTS + m\angle STW = 180^\circ \)). But if we look at the dropdown options, for the second statement (\( \angle TWV \cong \angle RTS \)), the reason is "Corresponding Angles Theorem".

So, if we are to fill the reason for the second statement:

To prove \( m\angle STW + m\angle TWV = 180^\circ \) given \( SU \parallel VX \):

• Statement 1: \( SU \parallel VX \) (Reason: Given).
• Statement 2: \( \angle TWV \cong \angle RTS \) (Reason: Corresponding Angles Theorem, as \( SU \parallel VX \) and \( RY \) is a transversal, so corresponding angles are congruent).
• Statement 3: \( m\angle RTS + m\angle STW = 180^\circ \) (Reason: Angles forming a linear pair sum to \( 180^\circ \)).
• Statement 4: \( m\angle STW + m\angle TWV = 180^\circ \) (Reason: Substitution, replacing \( \angle RTS \) with \( \angle TWV \) as they are congruent).

For the second statement's reason, from the dropdown, the correct option is "Corresponding Angles Theorem".

Answer:

Corresponding Angles Theorem