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Special Quadrilaterals on the GRE

Today, we’ll cover special quadrilaterals and the facts that you should know. These will allow you to answer questions quickly and understand the basic nature of these shapes.

 

1) A Typical Parallelogram

sq_img1

All parallelograms have the BIG FOUR properties:

1. Opposite sides are parallel—AB // CD, AD // BC
2. Opposite angles are congruent—∠BAD = ∠BCD, ∠ABC = ∠CDA
3. Opposite sides are congruent—AB = CD, AD = BC
4. Diagonals bisect each other—BE = ED, AE = EC

Additionally,

  • All parallelograms have all four of those properties
  • Any quadrilateral with any one of those properties has to be a parallelogram and has to have the other three.

 

2) A Typical Rectangle

A rectangle is a special kind of parallelogram.

sq_img2

  • All rectangles have the BIG FOUR parallelogram properties.
  • Equiangular quadrilateral—that is, four 90° angles—only the rectangle has this property.
  • The bisecting diagonals are also congruent—FH = GJ
  • Having the BIG FOUR plus congruent diagonals guarantees a shape is a rectangle.

NOTE: irregular quadrilateral can have congruent diagonals if the diagonals do not bisect each other.

 

3) A Typical Rhombus

A rhombus is a special kind of parallelogram.

sq_img3

  • All rhombuses have the BIG FOUR parallelogram properties.
  • Equilateral quadrilateral — that is, four each sides—LM = MN = NP = PL—only the rhombus has this property.
  • The bisecting diagonals are also perpendicular—MQN = 90°
  • Having the BIG FOUR plus perpendicular diagonals guarantees a shape is a rhombus.

NOTE: irregular quadrilateral can have perpendicular diagonals if the diagonals do not bisect each other.

 

4) The Square

There is an infinity of possible parallelogram shapes.

There is an infinity of possible rectangle shapes.

There is an infinity of possible rhombus shapes.

There is only one possible square shape—we can make that shape bigger or smaller, but the shape stays the same.

A square is a special kind of parallelogram, a special kind of rectangle, and a special kind of rhombus. It is the only rectangle that is also a rhombus, and vice versa.

A square has all parallelogram properties, all rectangle properties, and all rhombus properties.

To prove that a shape is a square, you would have to be able to prove that it is a parallelogram, and prove that it is a rectangle and prove that it is a rhombus.

 

 

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