Question

statement reason
1 ( overline{xy} cong overline{uv} ) given
2 ( angle wvx cong angle tuy ) given
3 ( angle tyu cong angle vxw ) given
4 ( uy = xy + ux ) additive property of length
5 ( vx = uv + ux ) additive property of length
6 ( uy = uv + ux ) substitution
7 ( vx = uy )
8 ( \triangle tuy cong \triangle wvx ) aas
9 ( overline{ty} cong overline{wx} ) additive property of angle measure
additive property of length
all right angles are congruent
alternate interior angles theorem
angles forming a linear pair sum to ( 180^{circ} )
asa
corresponding angles theorem
cpctc
definition of angle bisector
definition of congruence
definition of equilateral triangle
definition of midpoint
given
properties of addition, subtraction, multiplication, and division
reflexive property of congruence
reflexive property of equality
sas
sss
substitution
transitive property of congruence
transitive property of equality
vertical angle theorem

Explanation:

Step1: Analyze the given equalities

We have \( UY = XY + UX\) (from statement 4) and \( XY\cong UV\) (statement 1). By substitution (since \(XY = UV\) due to congruence), we can replace \(XY\) in the equation \(UY = XY + UX\) with \(UV\).

Step2: Substitute and get the result

After substitution, we get \(UY=UV + UX\). Also, from statement 5, \(VX = UV+UX\). So, by the transitive property of equality (if \(a = b\) and \(b = c\), then \(a = c\)), since \(UY=UV + UX\) and \(VX = UV + UX\), we can conclude that \(VX = UY\).

Answer:

Transitive Property of Equality