Question

statements
1 (overline{yw}) bisects (overline{ux}), (overline{uv} parallel overline{yw})
2 ?
3 ?
4 ?
5 (\triangle uvy cong \triangle wyx)

reasons
?
?
given
definition of a bisector
corresponding angles theorem
?

(overline{uy} cong overline{yx})
given
(angle u cong angle wyx)
asa
(angle x cong angle yvu)

Explanation:

Step1: Analyze Statement 1

Statement 1: \(\overline{YW}\) bisects \(\overline{UX}\), \(\overline{UV} \parallel \overline{YW}\). The reason should relate to the bisector and parallel lines. Since \(\overline{YW}\) bisects \(\overline{UX}\), by definition of a bisector, \(\overline{UY} \cong \overline{YX}\), but wait, the given reason for \(\overline{UY} \cong \overline{YX}\) is "Given"? No, wait, let's re - examine. Wait, the first statement's reason: since \(\overline{YW}\) bisects \(\overline{UX}\), the definition of a bisector (a bisector divides a segment into two congruent parts) would give \(\overline{UY} \cong \overline{YX}\), but in the table, \(\overline{UY} \cong \overline{YX}\) has reason "Given"? Maybe I misread. Wait, the problem is to fill the proof table. Let's start with statement 1: \(\overline{YW}\) bisects \(\overline{UX}\), \(\overline{UV} \parallel \overline{YW}\). The reason for statement 1 (if it's a given) would be "Given", but the table has "?" for the reason of statement 1. Wait, maybe the first step:

Step2: Statement 2

Reason for statement 2 is "Given". So statement 2 is a given fact. Let's assume statement 2 is something like \(\angle U \cong \angle WYX\) (since later we have \(\angle U \cong \angle WYX\) with reason "Given")? Wait, no, the "Given" in reason for statement 3? Wait, the table:

Statements:

1. \(\overline{YW}\) bisects \(\overline{UX}\), \(\overline{UV} \parallel \overline{YW}\)

2.?
3.?
4.?

1. \(\triangle UVY \cong \triangle WYX\)

Reasons:
1.?

1. Given
2. Definition of a Bisector
3. Corresponding Angles Theorem

5.?

And the lower part has:
\(\overline{UY} \cong \overline{YX}\) (Reason: Given)
\(\angle U \cong \angle WYX\) (Reason: Given)
ASA (for triangle congruence)
\(\angle X \cong \angle VYU\)

Let's build the proof:

1. Statement 1: \(\overline{YW}\) bisects \(\overline{UX}\), \(\overline{UV} \parallel \overline{YW}\)

• Reason 1: Given (since these are the initial conditions)

1. Statement 2: \(\angle U \cong \angle WYX\)

• Reason 2: Given (as per the lower table where \(\angle U \cong \angle WYX\) has reason "Given")

1. Statement 3: \(\overline{UY} \cong \overline{YX}\)

• Reason 3: Definition of a Bisector (because \(\overline{YW}\) bisects \(\overline{UX}\), so it divides \(\overline{UX}\) into two congruent segments \(\overline{UY}\) and \(\overline{YX}\))

1. Statement 4: \(\angle X \cong \angle VYU\)

• Reason 4: Corresponding Angles Theorem (since \(\overline{UV} \parallel \overline{YW}\), the alternate - interior or corresponding angles \(\angle X\) and \(\angle VYU\) are congruent)

1. Statement 5: \(\triangle UVY \cong \triangle WYX\)

• Reason 5: ASA (Angle - Side - Angle: we have \(\angle U \cong \angle WYX\), \(\overline{UY} \cong \overline{YX}\), and \(\angle X \cong \angle VYU\), so by ASA congruence criterion, \(\triangle UVY \cong \triangle WYX\))

Answer:

1. Statement: \(\overline{YW}\) bisects \(\overline{UX}\), \(\overline{UV} \parallel \overline{YW}\); Reason: Given
2. Statement: \(\angle U \cong \angle WYX\); Reason: Given
3. Statement: \(\overline{UY} \cong \overline{YX}\); Reason: Definition of a Bisector
4. Statement: \(\angle X \cong \angle VYU\); Reason: Corresponding Angles Theorem
5. Statement: \(\triangle UVY \cong \triangle WYX\); Reason: ASA