I'm confused with this question:

Question

The diagonals AC and BD of quadrilateral ABCD intersect at O. Given the information AO = BO and CO = DO, what can you deduce about the lengths of the sides of the quadrilateral? Prove your response.

Answer 1

ΔAOB∼ΔCOD by SAS similarity.

That means ∠BAO ≅ ∠DCA which by alternate interior angles means the line through AB // to line through CD.

ΔAOD ≅ ΔBOC by SAS, so AD=BC (CPCTE).

Based on the above it seems we have an isosceles trapezoid, which will be the case as long as AO ≠ DO. If AO = DO the quadrilateral becomes a rectangle. And if the diagonals are perpendicular the rectangle becomes a square.

I modified a Desmos graph I was using with another student to conform to this problem.

desmos.com/calculator/weqhjhd5gz

Use the sliders on a and b to modify the diagonal lengths and on the angles to modify angle of diagonals.

Answer 2

ΔAOD ≅ ΔBOC, SAS

AD ≅ BC, CPCFC

Now what can you prove about AB and CD?