Question

complete the proof that $\triangle suw\cong\triangle svt$.

statement	reason
2 $\overline{su}\cong\overline{sv}$	given
3 $\angle vsw\cong\angle tsu$	given
-4 $m\angle usw=m\angle usv + m\angle vsw$	additive property of angle measure
5 $m\angle tsv=m\angle tsu + m\angle usv$	additive property of angle measure
6 $m\angle usw=m\angle usv + m\angle tsu$	substitution
7

Explanation:

Step1: Use substitution

Since \(m\angle USW=m\angle USV + m\angle VSW\) (Step 4) and \(\angle VSW\cong\angle TSU\) (Step 3, so \(m\angle VSW = m\angle TSU\)), by substitution \(m\angle USW=m\angle USV + m\angle TSU\) (Step 6). Also, \(m\angle TSV=m\angle TSU + m\angle USV\) (Step 5). So, \(m\angle USW=m\angle TSV\), which means \(\angle USW\cong\angle TSV\).

Step2: Apply ASA (Angle - Side - Angle) congruence criterion

We know that \(\angle T\cong\angle W\) (Step 1), \(\overline{SU}\cong\overline{SV}\) (Step 2), and \(\angle USW\cong\angle TSV\) (from Step 1 result).
By the ASA (Angle - Side - Angle) congruence criterion for triangles (\(\triangle SUW\) and \(\triangle SVT\): \(\angle T\) in \(\triangle SVT\) and \(\angle W\) in \(\triangle SUW\), \(\overline{SU}\) in \(\triangle SUW\) and \(\overline{SV}\) in \(\triangle SVT\), \(\angle TSV\) in \(\triangle SVT\) and \(\angle USW\) in \(\triangle SUW\)).

Answer:

\(\triangle SUW\cong\triangle SVT\) by the ASA (Angle - Side - Angle) congruence criterion.