Question

complete the proof that \\( \triangle tuw \cong \triangle rxv \\).

statement	reason
2	\\( \overline{sv} \cong \overline{sw} \\)	given
3	\\( \overline{wx} \cong \overline{uv} \\)	given
4	\\( \overline{tu} \cong \overline{rx} \\)	given
5	\\( rv = rs + sv \\)	additive property of length
6	\\( tw = st + sw \\)
7	\\( rv = st + sw \\)	substitution
8	\\( rv = tw \\)	transitive property of equality
9	\\( vx = wx + vw \\)
10	\\( uw = uv + vw \\)	additive property of length
11	\\( vx = uv + vw \\)	substitution
12	\\( uw = vx \\)	transitive property of equality
13	\\( \triangle tuw \cong \triangle rxv \\)	sss

Explanation:

Step1: Determine the reason for statement 6

Since \(TW\) is composed of \(ST\) and \(SW\), by the Additive Property of Length, we have \(TW = ST+SW\).

Step2: Determine the reason for statement 9

Since \(VX\) is composed of \(WX\) and \(VW\), by the Additive Property of Length, we have \(VX = WX + VW\).

Answer:

For statement 6, the reason is "Additive Property of Length". For statement 9, the reason is "Additive Property of Length".