Existence of rhombi Squares do not necessarily exist in neut

Existence of rhombi. Squares do not necessarily exist in neutral geometry , but rhumbi exist in profusion. Prove this by showing that is line seg. AB and line seg. CD are two segments that share a common midpoint and line AB is perpendicular to line CD, then quadrilateral ACBD is a rhombus.

Solution

Let the common midpoint be E.
AE EB Definition of bisector
DE EB Definition of bisector
m<AEC m<DAB Reason: Vertical angles
AEC DAB SAS
AB DC Corresponding parts
Similarly,
m<AED m<CEB Reason: Vertical angles
AED CEB SAS
AD CB Corresponding parts
AB = AB Reflexive property
ABC ADC SSS
AB AD Corresponding parts
CB CD Corresponding parts
AB = BC = BD = AD Parts equal to equal parts are equal
ABCD is a rhombus. All sides are equal. Definiftion of a rhombus