Question

statement
1
2
3
4
5
6
7
8
reason

Explanation:

Step1: Given

From the problem statement, we are given that \(\overline{VW}\cong\overline{XY}\). This is the initial given information for the geometric proof.

Step2: Vertical Angles Theorem

\(\angle UXV\) and \(\angle SWY\) are vertical angles. By the Vertical Angles Theorem, vertical angles are congruent. So, \(\angle UXV\cong\angle SWY\).

Step3: Given

We are given that \(\angle U\cong\angle S\). This is another piece of given information for the proof.

Step4: Segment Addition Postulate

For the segment \(WY\), by the Segment Addition Postulate (if \(X\) is a point on segment \(WY\), then \(WY = XY+WX\)).

Step5: Segment Addition Postulate

For the segment \(VX\), by the Segment Addition Postulate (if \(W\) is a point on segment \(VX\), then \(VX = VW + WX\)).

Step6: Substitution Property

Since \(XY\cong VW\) (from step 1), we can substitute \(XY\) with \(VW\) in the equation \(WY=XY + WX\) (from step 4). So, \(WY=VW + WX\).

Step7: Transitive Property

We have \(VX = VW+WX\) (from step 5) and \(WY = VW + WX\) (from step 6). By the Transitive Property of equality (if \(a=b\) and \(c = b\), then \(a=c\)), \(VX=WY\).

Step8: Angle - Side - Angle (ASA) Congruence Criterion

We have \(\angle U\cong\angle S\) (step 3), \(VX\cong WY\) (step 7), and \(\angle UXV\cong\angle SWY\) (step 2). By the ASA (Angle - Side - Angle) congruence criterion for triangles, \(\triangle SWY\cong\triangle UXV\).

Answer:

1. Given
2. Vertical Angles Theorem
3. Given
4. Segment Addition Postulate
5. Segment Addition Postulate
6. Substitution Property
7. Transitive Property
8. ASA (Angle - Side - Angle) Congruence Criterion