Question

v is the midpoint of \\(\overline{rt}\\) and \\(\overline{su}\\). complete the proof that \\(\overline{tu} \parallel \overline{rs}\\).
(image of a quadrilateral with diagonals intersecting at v)

statement	reason
2 v is the midpoint of \\(\overline{su}\\)	given
3 \\(\overline{rv} \cong \overline{tv}\\)	definition of midpoint
4 \\(\overline{sv} \cong \overline{uv}\\)	definition of midpoint
5 \\(\angle rvs \cong \angle tvu\\)	vertical angle theorem
6 \\(\triangle rsv \cong \triangle tuv\\)	sas
7 \\(\angle rsv \cong \angle tuv\\)	cpctc
8 \\(\boxed{}\\)

Explanation:

Step1: Identify the goal

We need to prove \(\overline{TU} \parallel \overline{RS}\). From step 7, we have \(\angle RSV \cong \angle TUV\). These are alternate interior angles.

Step2: Apply the Alternate Interior Angles Theorem

If alternate interior angles are congruent, then the lines cut by the transversal are parallel. Here, the transversal is \(\overline{SU}\) (or \(\overline{SV}\) and \(\overline{UV}\)) cutting \(\overline{RS}\) and \(\overline{TU}\). Since \(\angle RSV \cong \angle TUV\) (alternate interior angles), we can conclude \(\overline{TU} \parallel \overline{RS}\) by the Alternate Interior Angles Theorem.

So the statement for step 8 is \(\overline{TU} \parallel \overline{RS}\) and the reason is "Alternate Interior Angles Theorem".

Answer:

Statement: \(\boldsymbol{\overline{TU} \parallel \overline{RS}}\)
Reason: Alternate Interior Angles Theorem