Is the converse of the rectangle diagonals conjecture true?

Answer 1

Converse: If the diagonals of a quadrilateral are congruent and bisect each other, then the

quadrilateral is a rectangle.

Given: Quadrilateral ABCD with diagonals , .

and _ bisect each other

Show: ABCD is a rectangle

Because the diagonals are congruent and bisect each other,

. Using the

Vertical Angles Theorem, AEB CED and BEC DEA. So ∆AEB ∆CED and ∆AED ∆CEB

by SAS. Using the Isosceles Triangle Theorem and CPCTC, 1 2 5 6, and 3 4 7

8. By the Angle Addition Postulate each angle of the quadrilateral is the sum of two angles, one

from each set. For example, mDAB = m1 + m8. By the addition property of equality, m1 m8

m2 m3 m5 m4 m6 m7. So by substitution mDAB mABC mBCD

mCDA. Therefore the quadrilateral is equiangular. Using 1 5 and the Converse of AIA, .

Using 3 7 and the Converse of AIA, . Therefore ABCD is an equiangular parallelogram,

so it is a rectangle by definition of rectangle.