Question

t is the midpoint of \\(\overline{qs}\\) and \\(\overline{pr}\\). complete the proof that \\(\overline{pq} \parallel \overline{rs}\\).\
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	statement	reason	\
---	---	---	\
1	t is the midpoint of \\(\overline{qs}\\)	given	\
2	t is the midpoint of \\(\overline{pr}\\)	given	\
3	\\(\overline{qt} \cong \overline{st}\\)	definition of midpoint	\
4	\\(\overline{pt} \cong \overline{rt}\\)	definition of midpoint	\
5	\\(\angle ptq \cong \angle rts\\)	vertical angle theorem	\
6	\\(\triangle pqt \cong \triangle rst\\)	sas	\
7	\\(\angle pqt \cong \angle rst\\)	cpctc	\
8

Explanation:

Step-by-Step Explanation:

Step 1: Identify the relationship from congruent angles

We know that \( \angle PQT \cong \angle RST \) (from step 7, CPCTC). These are alternate interior angles formed by the transversal \( QS \) intersecting lines \( PQ \) and \( RS \).

Step 2: Apply the Alternate Interior Angles Theorem

The Alternate Interior Angles Theorem states that if two lines are cut by a transversal and the alternate interior angles are congruent, then the two lines are parallel. Since \( \angle PQT \) and \( \angle RST \) are alternate interior angles and they are congruent, we can conclude that \( \overline{PQ} \parallel \overline{RS} \).

Answer:

Statement 8: \( \overline{PQ} \parallel \overline{RS} \)
Reason 8: Alternate Interior Angles Theorem (or Converse of Alternate Interior Angles Theorem)