Question

statement reason
1 ( overline{xy} cong overline{uv} ) given
2 ( overline{vw} cong overline{tu} ) given
3 ( overline{ty} cong overline{wx} ) given
4 ( uy = xy + ux )
5 ( vx = uv + ux )
6 ( uy = uv + ux ) substitution
7 ( vx = uy )
8 ( \triangle vwx cong \triangle uty ) sss

Explanation:

Step1: Segment addition postulate

$UY = XY + UX$ (by segment addition postulate which states that if \(X\), \(Y\), \(U\) are collinear points with \(X\) between \(U\) and \(Y\), then the length of \(UY\) is the sum of the lengths of \(UX\) and \(XY\))

Step2: Segment addition postulate

$VX=UV + UX$ (by segment addition postulate which states that if \(U\), \(V\), \(X\) are collinear points with \(U\) between \(V\) and \(X\), then the length of \(VX\) is the sum of the lengths of \(UV\) and \(UX\))

Step3: Substitution

Since \(XY\cong UV\) (given \(XY = UV\)), substitute \(XY\) with \(UV\) in \(UY = XY+UX\) to get \(UY = UV + UX\)

Step4: Transitive property

Since \(VX=UV + UX\) (from step 2) and \(UY = UV + UX\) (from step 3), then \(VX = UY\) (if \(a=b\) and \(c = b\), then \(a=c\))

Answer:

1. Segment Addition Postulate; 5. Segment Addition Postulate; 7. Transitive Property of Equality