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embizze | Certified Educator

The quadrilateral shown has diagonals that bisect each other, and the diagonals are congruent.

Since the diagonals bisect each other the quadrilateral must be a parallelogram.

** This can be seen since there are two sets of congruent triangles (using SAS with the vertical angles); marking the corresponding angles congruent we see that the opposite sides are cut by diagonals forming congruent alternate interior angles. Thus the opposite sides are parallel. **

Since the diagonals are congruent the parallelogram must be a rectangle.

** This can be seen since the triangles formed are isosceles. Thus all of the angles of the parallelogram are congruent making this a rectangle. **

We cannot conclude that this is a square as there is no way to show that all four sides are congruent. (If the diagonals were perpendicular then the quadrilateral would be a rhombus, and with the other given information would be a square. But this is not given.)

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