Properties and Proofs of Quadrilaterals

Vocabulary flashcards detailing the properties and methods of proof for common quadrilaterals based on lecture notes.

Quadrilateral

A closed four-sided figure whose interior angles add up to $360^\text{o}$.

Square (Properties)

A quadrilateral in which all sides are equal in length, opposite sides are parallel, corner angles equal $90^\text{o}$, diagonals are equal and bisect each other at $90^\text{o}$, and diagonals bisect the corner angles.

Rectangle (Properties)

A quadrilateral in which opposite sides are parallel and equal in length, corner angles equal $90^\text{o}$, and diagonals are equal and bisect each other.

Parallelogram (Parm) (Properties)

A quadrilateral in which opposite sides are parallel and equal in length, opposite angles are equal, and diagonals bisect each other.

Rhombus (Properties)

A quadrilateral in which all sides are equal in length, opposite sides are parallel, opposite angles are equal, diagonals bisect each other at $90^\text{o}$, and diagonals bisect the corner angles.

Kite (Properties)

A quadrilateral with two pairs of adjacent sides equal, a single pair of opposite angles equal, diagonals intersecting at $90^\text{o}$, one diagonal bisecting the corner angles, and the shorter diagonal bisected by the longer diagonal.

Trapezium (Properties)

A quadrilateral in which one pair of opposite sides are parallel.

Proving a Parallelogram (parm)

To prove a quadrilateral is a parallelogram, prove any one of the following: 1) Both pairs of opposite sides parallel; 2) Both pairs of opposite sides equal; 3) Both pairs of opposite angles equal; 4) One pair of opposite sides equal and parallel; 5) Diagonals bisect each other (they have the same midpoint).

Proving a Rhombus

Prove the quadrilateral is a parallelogram and one of the following: 1) One pair of adjacent sides equal; 2) Diagonals bisect at $90^\text{o}$.

Proving a Square

Prove any one of the following: 1) The quadrilateral is a parm with a corner angle equal to $90^\text{o}$ and a pair of adjacent sides equal; 2) The quadrilateral is a rhombus with a corner angle equal to $90^\text{o}$; 3) The quadrilateral is a rhombus with equal diagonals.

Proving a Rectangle

Prove the quadrilateral is a parallelogram and one of the following: 1) The parm has a corner angle equal to $90^\text{o}$; 2) The parm has diagonals that are equal in length.

Proving a Kite

Prove that each pair of adjacent sides is equal in length.

Proving a Trapezium

Prove that one pair of opposite sides are parallel.

Quadrilateral

A closed four-sided figure whose interior angles add up to $360^\circ$.

Square (Properties)

A quadrilateral in which all sides are equal in length, opposite sides are parallel, corner angles equal $90^\circ$, diagonals are equal and bisect each other at $90^\circ$, and diagonals bisect the corner angles.

Rectangle (Properties)

A quadrilateral in which opposite sides are parallel and equal in length, corner angles equal $90^\circ$, and diagonals are equal and bisect each other.

Parallelogram (Parm) (Properties)

A quadrilateral in which opposite sides are parallel and equal in length, opposite angles are equal, and diagonals bisect each other.

Rhombus (Properties)

A quadrilateral in which all sides are equal in length, opposite sides are parallel, opposite angles are equal, diagonals bisect each other at $90^\circ$, and diagonals bisect the corner angles.

Kite (Properties)

A quadrilateral with two pairs of adjacent sides equal, a single pair of opposite angles equal, diagonals intersecting at $90^\circ$, one diagonal bisecting corner angles, and the shorter diagonal bisected by the longer diagonal.

Trapezium (Properties)

A quadrilateral with one pair of opposite sides parallel.

Proving a Parallelogram (Parm)

Prove any one of the following: 1) Both pairs of opposite sides parallel; 2) Both pairs of opposite sides equal; 3) Both pairs of opposite angles equal; 4) One pair of opposite sides equal and parallel; 5) Diagonals bisect each other (same midpoint).

Proving a Rhombus

Prove the quadrilateral is a parallelogram and one of the following: 1) One pair of adjacent sides equal; 2) Diagonals bisect at $90^\circ$.

Proving a Square

Prove any one of the following: 1) The quadrilateral is a parm with a corner angle equal to $90^\circ$ and a pair of adjacent sides equal; 2) The quadrilateral is a rhombus with a corner angle equal to $90^\circ$; 3) The quadrilateral is a rhombus with equal diagonals.

Proving a Rectangle

Prove the quadrilateral is a parallelogram and one of the following: 1) The parm has a corner angle equal to $90^\circ$; 2) The parm has diagonals that are equal in length.

Proving a Kite

Prove that each pair of adjacent sides is equal in length.

Proving a Trapezium

Prove that one pair of opposite sides are parallel.