Question

in the diagram, \\( \overline { d g } \parallel \overline { e f } \\).
what additional information would prove that defg is an
isosceles trapezoid?
\\( \bigcirc \overline { d e } \cong \overline { g f } \\)
\\( \bigcirc \overline { d e } \cong \overline { d g } \\)
\\( \bigcirc \overline { e f } \cong \overline { d g } \\)
\\( \bigcirc \overline { e f } \cong \overline { g f } \\)

Explanation:

Step1: Recall the definition of an isosceles trapezoid

An isosceles trapezoid is a trapezoid (a quadrilateral with one pair of parallel sides) where the non - parallel sides (the legs) are congruent. In trapezoid \(DEFG\) with \(DG\parallel EF\), the non - parallel sides are \(DE\) and \(GF\).

Step2: Check the congruence of non - parallel sides

We need to check which option gives the congruence of the non - parallel sides.

• For \(\overline{DE}\cong\overline{GF}\), since \(DG\parallel EF\) and if the non - parallel sides \(DE\) and \(GF\) are congruent, by the definition of an isosceles trapezoid, \(DEFG\) is an isosceles trapezoid.
• For \(\overline{DE}\cong\overline{DG}\), \(DE\) and \(DG\) are not the non - parallel sides with respect to the parallel sides \(DG\parallel EF\).
• For \(\overline{EF}\cong\overline{DG}\), \(EF\) and \(DG\) are the parallel sides. Congruence of parallel sides does not make it an isosceles trapezoid.
• For \(\overline{EF}\cong\overline{GF}\), \(EF\) and \(GF\) are not the non - parallel sides with respect to the parallel sides \(DG\parallel EF\).

Answer:

\(\overline{DE}\cong\overline{GF}\) (first option)