Question

given: quadrilateral lprs,
\\( \overline { pt } \parallel \overline { rs } \\), and
\\( \overline { pt } \cong \overline { rs } \\)
prove: lprs is a trapezoid.

1. quadrilateral lprs,

\\( \overline { pt } \parallel \overline { rs } \\), and \\( \overline { pt } \cong \overline { rs } \\)

1. prst is a parallelogram.
2. \\( \angle ltp \cong \angle rpt \\)
3. given
4. if one pair of opposite sides is both parallel and congruent, then the quadrilateral is a parallelogram.
5. transitive property

Explanation:

Step1: Analyze the property of parallelogram

Since \(PRST\) is a parallelogram (from statement 2), \(PR\parallel ST\).

Step2: Recall the definition of trapezoid

A trapezoid is a quadrilateral with one pair of parallel sides. In quadrilateral \(LPRS\), we have \(PR\parallel LS\) (because \(ST\) is a part of \(LS\) and \(PR\parallel ST\)).

Answer:

Since \(PRST\) is a parallelogram (so \(PR\parallel ST\)), and \(ST\) is on \(LS\), in quadrilateral \(LPRS\) there is one pair of parallel sides (\(PR\) and \(LS\)). By the definition of a trapezoid (a quadrilateral with one pair of parallel sides), \(LPRS\) is a trapezoid.