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.

Explanation:

Step1: Recall the definition of a parallelogram

A quadrilateral with one pair of opposite sides parallel and congruent is a parallelogram. Since \( \overline{PT}\parallel\overline{RS}\) and \( \overline{PT}\cong\overline{RS}\), by the parallelogram definition, \(PRST\) is a parallelogram.

Step2: Use the property of a parallelogram

In a parallelogram \(PRST\), \( \overline{PR}\parallel\overline{TS}\) (opposite sides of a parallelogram are parallel).

Step3: Recall the definition of a trapezoid

A trapezoid is a quadrilateral with at least one pair of parallel sides. In quadrilateral \(LPRS\), \( \overline{PR}\parallel\overline{TS}\) (from step 2), so \(LPRS\) is a trapezoid.

Answer:

1. A quadrilateral with one pair of opposite sides parallel and congruent is a parallelogram.
2. Opposite sides of a parallelogram are parallel. A trapezoid is a quadrilateral with at least one pair of parallel sides.