Question

$\angle u \cong \angle x$, $\overline{uv} \cong \overline{wx}$, and $\angle twu \cong \angle rvx$. complete the proof that $\triangle rvx \cong \triangle twu$.

(image of triangles with points x, w, v, u and r, s, t)

	statement	reason
2	$\overline{uv} \cong \overline{wx}$	given
3	$\angle twu \cong \angle rvx$	given
4	$vx = wx + vw$	additive property of length
5	$uw = uv + vw$	additive property of length
6	$vx = uv + vw$
7	$uw = vx$
8	$\triangle rvx \cong \triangle twu$	asa

Explanation:

Step1: Analyze Statement 6

We know from statement 2 that $\overline{UV} \cong \overline{WX}$, which means $UV = WX$ (since congruent segments have equal length). From statement 4, $VX = WX + VW$. Substituting $UV$ for $WX$ (because $UV = WX$), we get $VX = UV + VW$. The reason for this substitution is the Substitution Property (since we replace $WX$ with its equal $UV$ in the equation from statement 4).

Step2: Analyze Statement 7

From statement 5, $UW = UV + VW$, and from statement 6, $VX = UV + VW$. By the Transitive Property of Equality (if $a = b$ and $b = c$, then $a = c$), we can conclude that $UW = VX$. So the reason for statement 7 is the Transitive Property of Equality (or Substitution, since both $UW$ and $VX$ equal $UV + VW$).

For statement 6:
The reason is "Substitution (since $UV = WX$ from $\overline{UV} \cong \overline{WX}$ and substituting into $VX = WX + VW$)".

For statement 7:
The reason is "Transitive Property of Equality (since $UW = UV + VW$ and $VX = UV + VW$, so $UW = VX$)".

Answer:

Statement 6 Reason: Substitution (using $\overline{UV} \cong \overline{WX}$ to replace $WX$ with $UV$ in $VX = WX + VW$)
Statement 7 Reason: Transitive Property of Equality (or Substitution, as both equal $UV + VW$)

(If we need to fill in the table cells:
Statement 6 Reason: Substitution (from $\overline{UV} \cong \overline{WX}$ so $UV = WX$)
Statement 7 Reason: Transitive Property of Equality (or Substitution, since $UW = UV + VW$ and $VX = UV + VW$))