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Determine whether the conjecture is true or false. Give a counterexample for any false conjecture.

Given: points A, B, C, and D
Conjecture: A, B, C, and D are coplanar.

A. False; the four points do not have to be in a straight line.
B. False; two points are always coplanar but four are not.
C. True
D. False; three points are always coplanar but four are not.

The correct answers are B and D.

A is false because, even though it's true that 4 points are not necessarily on the same line, it is not required that they are collinear in order for them to be coplanar.

B is true: two points always lie on the same line (and thus also in infinite same planes), but four points are not always coplanar

C is false. As a counterexample pick the four points (0,0, 0), (1, 0, 0), (0, 1, 0), (0, 0, 1).

D is true: two points are always on the same line, three points always lie in a common plane, but 4 points are not necessarily coplanar.