Question

use the given information to prove that \\(overline{ef} \cong overline{gd}\\).
given: \\(overline{ef} \parallel overline{gd}\\), \\(overline{fg} \parallel overline{de}\\)
prove: \\(overline{ef} \cong overline{gd}\\)

Explanation:

Step1: Identify the figure type

The figure is a parallelogram (since \( EF \parallel GD \) and \( FG \parallel DE \), as given by the parallel line statements). In a parallelogram, opposite sides are congruent.

Step2: Apply parallelogram property

In parallelogram \( DEFG \) (assuming the vertices are \( D, E, F, G \) in order), \( EF \) and \( GD \) are opposite sides. By the property of parallelograms, opposite sides are congruent, so \( \overline{EF} \cong \overline{GD} \).

Step3: Justify the proof steps

First, confirm the figure is a parallelogram using the given parallel lines (\( EF \parallel GD \) and \( FG \parallel DE \)). Then, use the parallelogram opposite sides congruence theorem to conclude \( EF \cong GD \).

Answer:

To prove \( \overline{EF} \cong \overline{GD} \):

1. Given: \( \overline{EF} \parallel \overline{GD} \) and \( \overline{FG} \parallel \overline{DE} \) (so \( DEFG \) is a parallelogram, as a quadrilateral with both pairs of opposite sides parallel is a parallelogram).
2. Property of Parallelograms: In a parallelogram, opposite sides are congruent. Thus, \( \overline{EF} \cong \overline{GD} \) (since \( EF \) and \( GD \) are opposite sides of parallelogram \( DEFG \)).

Final conclusion: \( \overline{EF} \cong \overline{GD} \) is proven by the property of parallelograms (opposite sides congruent) after confirming \( DEFG \) is a parallelogram from the given parallel sides.