**Quadrilaterals: ****GMAT / GRE Mathematics Tutorial**

This tutorial is based on geometry concepts. We will study the types and properties of five quadrilaterals: Parallelogram, rectangle, square, rhombus and trapezium.

**Basic definition**

In Euclidean plane geometry, a quadrilateral is basically a four sided polygon, with the internal angles adding up to 360°.

**Types of quadrilaterals**

Quadrilaterals are of five types based on their shapes. These are:

- Parallelogram
- Rectangle
- Square
- Rhombus
- Trapezium

We’ll study each of these quadrilaterals in detail.

**Parallelogram**

A parallelogram is a quadrilateral with opposite sides parallel (as seen in the figure below).

**Properties of a parallelogram**

- Opposite angles are equal.
- Opposite sides/facing sides are parallel and of equal length.
- Adjacent angles are supplementary, i.e. they add up to 180°.
- Each diagonal divides the parallelogram into two equal triangles. The two diagonals bisect each other.
- If one of the angles of a parallelogram is a right angle then the parallelogram is a rectangle.

**Important formulas**

If the length of a parallelogram is ‘l’, breadth is ‘b’ and height is ‘h’ then:

- Area = L * H
- Perimeter = 2(L+B)

**Rectangles**

A rectangle is a quadrilateral with all angles at right angles (360°/4 = 90°). (As seen in the figure below)

**Properties of a Rectangle**

- Opposite sides are both parallel and equal to each other.
- All angles are 90°.
- The diagonals are equal and bisect each other (divide each other equally).
- Opposite angles formed at the point where diagonals meet are equal.
- A rectangle is a parallelogram with all the angles 90°.

**Important formulas**

- If L = length and B = breadth, then

Length of the diagonal = √ (L^{2} + B^{2})

- Area = L * B
- Perimeter = 2(L+B)

**Squares**

A square is a quadrilateral in which all the sides are equal (as seen in the figure above).

**Properties of a square**

- All sides and angles are equal.
- Opposite sides are parallel to each other.
- The diagonals are equal and bisect each other at right angles.
- A square is a type of parallelogram in which all the angles and sides are equal.
- Also, a parallelogram becomes a square when the diagonals are equal and right bisectors of each other.

**Important formulas for Squares**

- If ‘L’ is the length of the side of a square then length of the diagonal = L √2.
- Area = L
^{2}. - Perimeter = 4L

## Rhombus

A rhombus is a quadrilateral having equal sides and opposite sides are parallel to each other. (As seen in the figure above)

**Properties of a Rhombus**

- All sides are equal
- Opposite angles are equal
- The diagonals bisect each other at right angles
- Adjacent angles are supplementary (e.g. ∠A + ∠B = 180°)
- A rhombus is also a parallelogram that has diagonals that are perpendicular to each other
- A rhombus with right angles is a square

**Important formulas for a Rhombus**

If a and b are the lengths of the diagonals of a rhombus,

- Area = (a* b) / 2
- Perimeter = 4L

**Trapezium**

A trapezium or a trapezoid is a quadrilateral that has only one pair of parallel sides called bases and two lateral sides called legs.

**Properties of a Trapezium**

- Only the bases of the trapezium are parallel to each other (MN ⫽ OP).
- None of the sides, angles and diagonals are equal.

**Important Formulas for a Trapezium**

- Area = (1/2) h (L+L
_{2}) - Perimeter = L + L
_{1}+ L_{2}+ L_{3}

**Summary of properties**

Summarizing what we have learned so far for quick reference and recollection:

S.No. | Property | Parallelogram | Rectangle | Rhombus | Square |
---|---|---|---|---|---|

1 | All sides are congruent | ✕ | ✕ | ✓ | ✓ |

2 | Opposite sides are parallel and congruent | ✓ | ✓ | ✓ | ✓ |

3 | All angles are congruent | ✕ | ✓ | ✕ | ✓ |

4 | Opposite angles are congruent | ✓ | ✓ | ✓ | ✓ |

5 | Diagonals are congruent | ✕ | ✓ | ✕ | ✓ |

6 | Diagonals are perpendicular | ✕ | ✕ | ✓ | ✓ |

7 | Diagonals bisect each other | ✓ | ✓ | ✓ | ✓ |

8 | Adjacent angles are supplementary | ✓ | ✓ | ✓ | ✓ |

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