Question

got it? 2. can you prove that the quadrilateral is a parallelogram based on the given information? explain.
a. given: \\( \overline{ef} \cong \overline{gd}, \overline{de} \parallel \overline{fg} \\) prove: \\( defg \\) is a parallelogram.
b. given: \\( \angle aln \cong \angle dnl, \angle anl \cong \angle dln \\) prove: \\( land \\) is a parallelogram.

Explanation:

Part a

Step1: Use the property of parallelogram

One of the properties of a parallelogram is that if one pair of opposite sides of a quadrilateral is both parallel and congruent, then the quadrilateral is a parallelogram.
In quadrilateral \(DEFG\), we are given that \(\overline{DE}\parallel\overline{FG}\) (one pair of opposite sides are parallel) and \(\overline{EF}\cong\overline{GD}\). But we need to check the pair of sides that are parallel.
Since \(\overline{DE}\parallel\overline{FG}\), we should check the congruence of \(\overline{DE}\) and \(\overline{FG}\) or \(\overline{GD}\) and \(\overline{EF}\) along with the parallel - side condition.
If we consider the pair of sides \(\overline{DE}\) and \(\overline{FG}\), we only know they are parallel. If we consider the pair of sides \(\overline{GD}\) and \(\overline{EF}\), we only know they are congruent.
We cannot prove that \(DEFG\) is a parallelogram based on the given information \(\overline{EF}\cong\overline{GD},\overline{DE}\parallel\overline{FG}\)

Part b

Step1: Use the Angle - Angle - Side (AAS) congruence criterion

In \(\triangle ALN\) and \(\triangle DNL\), we know that \(\angle ALN\cong\angle DNL\), \(\angle ANL\cong\angle DLN\), and \(LN = LN\) (common side).
By the AAS (Angle - Angle - Side) congruence criterion, \(\triangle ALN\cong\triangle DNL\)

Step2: Use the property of congruent triangles

Since \(\triangle ALN\cong\triangle DNL\), then \(AL = DN\) and \(AN=DL\) (corresponding parts of congruent triangles are congruent)

Step3: Use the parallelogram criterion

If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram.
In quadrilateral \(LAND\), since \(AL = DN\) and \(AN = DL\), quadrilateral \(LAND\) is a parallelogram

Answer:

a. No. The given information (\(\overline{EF}\cong\overline{GD},\overline{DE}\parallel\overline{FG}\)) does not satisfy the criteria (a pair of opposite sides that are both parallel and congruent) for a parallelogram.
b. Yes. Since \(\triangle ALN\cong\triangle DNL\) (by AAS), \(AL = DN\) and \(AN = DL\). By the property that if both pairs of opposite sides of a quadrilateral are congruent, the quadrilateral is a parallelogram, \(LAND\) is a parallelogram.