In quadrilateral ABCD the points B and C lie on the same side of the line AD, and measurement gives ∠ABD = 40° and ∠ACD = 40°. What follows from this?

The segment AD subtends the same angle, 40°, at B and at C, and B and C lie on the same side of AD. The locus of points on one side of AD from which AD subtends a fixed angle is a single arc through A and D, so B and C both lie on that arc — hence all four points lie on one circle. (This is the converse of the equal-angles-in-the-same-segment theorem.) The other options need extra information: nothing forces AB = CD or BC ∥ AD, and AD would be a diameter only if the subtended angle were 90°, not 40°.