Why is a square always a rhombus, but a rhombus is not always a square?

Question

Ella Armstrong · Accepted Answer

It is important to work with definitions first.

A parallelogram is a quadrilateral with two pairs of opposite sides parallel. A rhombus is a parallelogram with equal sides A square is a rhombus with all the angles equal (to 90°). Students often make the mistake of defining a rhombus as "A rhombus is a square pushed over." It would be better to say that a square is a rhombus pushed up straight. In a #color(blue)"rhombus"# #color(blue)"All the sides are equal."# #color(blue)"The opposite sides are parallel"# #color(blue)"The opposite angles are equal"# #color(blue)"The diagonals bisect each other at 90°"# #color(blue)"The diagonals bisect the angles at the vertices"# #color(blue)"there are 2 lines of symmetry"# #color(blue)"it has rotational symmetry of order 2"# A square has all the properties of a rhombus, with more properties - In a #color(red)"square:"# #color(blue)"All the sides are equal."# #color(blue)"The opposite sides are parallel"# #color(blue)"The opposite angles are equal"# #color(red)"All the angles are equal to 90°."# #color(blue)"The diagonals bisect each other at 90°"# #color(red)"The diagonals are equal."# #color(red)"The diagonals bisect the angles to give 45° angles"# #color(red)"there are 4 lines of symmetry"# #color(red)"it has rotational symmetry of order 4"# A rhombus does NOT have all the properties of a square, therefore is not a special kind of square.

Savannah Aguilar · Answer

undefined A square is always a rhombus because it meets the definition of a rhombus, which is a quadrilateral with all sides of equal length. However, a rhombus is not always a square because it does not necessarily have all interior angles of 90 degrees, which is a defining characteristic of a square.