Question

1. complete each statement for rhombus uvwx.

a. (overline{uv} cong __ cong cong __)
b. (angle u w cong angle __) and (angle x u v cong angle __)
c. (overline{u v} | __) and (overline{u x} | __)
d. (overline{u e} cong __) and (overline{x e} cong __)

Explanation:

Part a

Step1: Recall the property of a rhombus

In a rhombus, all four sides are congruent.

Step2: Identify the sides

Since \(UVWX\) is a rhombus, \(\overline{UV}\cong\overline{VW}\cong\overline{WX}\cong\overline{XU}\)

Part b

Step1: Recall the property of alternate - interior angles in a rhombus (parallel sides)

In a rhombus \(UVWX\), \(UX\parallel VW\) and \(UV\parallel XW\). The diagonals bisect the angles.
Since \(UX\parallel VW\) and \(UW\) is a transversal, \(\angle UWW\cong\angle VWU\) (alternate - interior angles).
Since \(UV\parallel XW\) and \(XV\) is a transversal, \(\angle XUV\cong\angle WXV\) (alternate - interior angles)

Part c

Step1: Recall the property of a rhombus (parallel sides)

In a rhombus, opposite sides are parallel.
So, \(\overline{UV}\parallel\overline{XW}\) and \(\overline{UX}\parallel\overline{VW}\)

Part d

Step1: Recall the property of the diagonals of a rhombus

The diagonals of a rhombus bisect each other.
So, \(\overline{UE}\cong\overline{WE}\) and \(\overline{XE}\cong\overline{VE}\)

Answer:

a. \(\overline{VW}\), \(\overline{WX}\), \(\overline{XU}\)
b. \(\angle VWU\), \(\angle WXV\)
c. \(\overline{XW}\), \(\overline{VW}\)
d. \(\overline{WE}\), \(\overline{VE}\)