(Reprinted with permission from Mathematics in the Middle, copyright 1998 by the National Council of Teachers of Mathematics. All rights reserved.)

Complete the conjectures below by writing always, sometimes, or never in the blanks provided. After completing the statements, return to each response and justify it by using the following criteria:

  • If you wrote “always,” provide three examples. Then try to come up with a proof!
  • If you wrote “sometimes,” provide two examples (one true and one false).
  • If you wrote “never,” provide a counterexample.

Quadrilaterals ________________ have at least two lines of symmetry.

The diagonals of a trapezoid divide the trapezoid into four triangles, _________ having the same area.

The diagonals of a rectangle __________ bisect each other.

The angle bisectors of a convex quadrilateral __________ meet at a point.

The perpendicular bisectors of the sides of a parallelogram _________ meet at a point.