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1. AC is perpendicular to BD 2. \(AB + CD = BC + AD\)

Answer is E. But as far as I learned in school, the ONLY 2 quadrilaterals that have perpendicular diagonals are the rhombus and the square and both have all sides equal. I do not find the explanation correct. _________________

Not sure if my reasoning is correct, but this is what I got.

Draw a kite shape, with A being the top tip of the kite, and C being the bottom tip; B and D are opposite side corners. AC and BD (where the kite frame sticks would go) are perpendicular. And since all perfect kites are symmetric, then the sides AB=AD, and BC=CD, so AB+CD=BC+AD. But obviously AB does not equal BC. Or you would have a square kite. Answer is E.

Could you please explain statement wise why they are insufficient?

If ABCD is a quadrilateral, is AB=BC=CD=DA ?

(1) AC is perpendicular to BD. The diagonals are perpendicular to each other: ABCD could be a kite (answer NO), a rhombus (answer YES) or a square, which is just a special type of rhombus (answer YES). Not sufficient.

(2) AB+CD=BC+DA. The sum of opposite sides are equal. Clearly insufficient.

(1)+(2) ABCD could be a kite (see the diagram below) - answer NO or a square/rhombus - answer YES. Not sufficient.