Questions: The figure below shows a quadrilateral ABCD such that AB and DC are equal and parallel. A student wrote the following sentences to prove that quadrilateral ABCD is a parallelogram: Quadrilateral ABCD is a parallelogram because its opposite sides are equal and parallel. Which statement best describes an error in the student's proof? Triangles ABO and BCD are congruent by the SSS postulate instead of the SAS postulate. Triangles ABO and BCD are congruent by the AAS postulate instead of the SAS postulate. Angle DBC and angle BDA form a pair of corresponding angles, not a pair of vertical angles. Angle DEC and angle BOA form a pair of alternate interior angles that are congruent, not a pair of vertical angles.

The figure shows quadrilateral ABCD where sides AB and DC are equal and parallel. A student claims ABCD is a parallelogram because its opposite sides are equal and parallel. Analyze the flaw in the student's reasoning.

Analyze the given information

We are given that AB and DC are equal and parallel. This is insufficient to conclude that ABCD is a parallelogram.

Consider the conditions for a parallelogram

A quadrilateral is a parallelogram if:

1. Both pairs of opposite sides are parallel.
2. Both pairs of opposite sides are equal.
3. One pair of opposite sides are equal and parallel.
4. Both pairs of opposite angles are equal.
5. Diagonals bisect each other. The student states the definition of a parallelogram but doesn't prove it. They only state that one pair of sides is both equal and parallel. They have not shown that the other pair of sides (BC and AD) are equal and/or parallel.

Consider the congruent triangles

If we try to prove the triangles ABO and CDO congruent, we have AB = CD and the parallel lines give us alternate interior angles being equal, i.e., angle ABD = angle CDB and angle BAC = angle DCA. However, we can't directly use SSS or SAS criteria here.

Identify the flaw

The student assumes that since one pair of sides is parallel and equal, then the quadrilateral is a parallelogram. The student needs to prove that both pairs of opposite sides are parallel. The student can prove that the triangles ABD and CDB are congruent by SAS postulate (AB=CD, BD=BD, and angle ABD = angle CDB). After showing these triangles are congruent, we can conclude that AD=BC and AD || BC.

\\(\boxed{\text{Triangles ABD and CDB are congruent by the SAS postulate instead of the SSS postulate}}\)

\\(\boxed{\text{Triangles ABD and CDB are congruent by the SAS postulate instead of the SSS postulate}}\)