Question

select the correct answer from each drop - down menu.
consider parallelogram abcd with center e.
since rotating parallelogram abcd about its center e maps side \\( \overline { cd } \\) onto side \\( \overline { ab } \\) and side \\( \overline { bc } \\) onto side, opposite sides of parallelograms are.

Explanation:

Step1: Properties of parallelogram rotation

A parallelogram has 180 - degree rotational symmetry about its center. When we rotate a parallelogram \(ABCD\) by \(180^{\circ}\) about its center \(E\), each point \(P(x,y)\) is mapped to \(P'(-x,-y)\) (in coordinate - plane terms, but more generally in geometric transformation).
For side \(\overline{CD}\), after a \(180^{\circ}\) rotation about \(E\), it maps to \(\overline{AB}\).

Step2: Mapping of side \(\overline{BC}\)

When we rotate side \(\overline{BC}\) by \(180^{\circ}\) about the center \(E\) of the parallelogram \(ABCD\), using the property of rotational symmetry of parallelograms. If we consider the vectors or the geometric positions of the vertices, a \(180^{\circ}\) rotation about the center of the parallelogram. Let \(B\) and \(C\) be two vertices of the parallelogram. After a \(180^{\circ}\) rotation about \(E\), the side \(\overline{BC}\) maps to \(\overline{AD}\).

Step3: Property of opposite sides of parallelograms

One of the fundamental properties of parallelograms is that opposite sides are congruent. In parallelogram \(ABCD\), \(AB = CD\) and \(BC=AD\)

Answer:

Since rotating parallelogram \(ABCD\) \(180^{\circ}\) about its center \(E\) maps side \(\overline{CD}\) onto side \(\overline{AB}\) and side \(\overline{BC}\) onto side \(\overline{AD}\), opposite sides of parallelograms are congruent.