Proving Quadrilaterals

Proving Quadrilaterals

Parallelogram

1. If both pairs of opposite sides of a quadrilateral are parallel, then it is a parallelogram (definition)

2. If both pairs of opposite angles or sides of a quadrilateral are congruent, then it is a parallelogram

3. If the diagonals of a quadrilateral bisect each other, then it is a parallelogram

Rectangle

In order to prove that it is a rectangle you must prove that it is a parallelogram first then use one of the following

2. If a parallelogram contains at least one right angle, then it is a rectangle(definition)

3. If the diagonals of a parallelogram are congruent, it is a rectangle

You can also show that it is a rectangle without first showing it is a parallelogram

4. If all four angles of a quadrilateral are right angles, it's a rectangle

Kites

1. If two disjoint pairs of consecutive sides of a quadrilateral are congruent, then it's a kite

2. If one of the diagonals of a quadrilateral is the perpendicular bisector of the other diagonal, then it's a kite

Rhombus

You must prove that it is a parallelogram first then prove it a rhombus by any of the following

1. if a parallelogram contains a pair of consecutive sides that are congruent, then it's a rhombus(definition)

2. If either diagonal of a parallelogram bisects two angles of the parallelogram, then it's a rhombus

You can also prove it a rhombus without proving it a parallelogram first

3. if the diagonals of a quadrilateral are perpendicular bisectors of each other, then it's a rhombus

Square

1. If a quadrilateral is both a rectangle and a rhombus, then it is a square

Trapezoid, Isosceles Trapezoid

1. If the nonparallel sides of a trapezoid are congruent then it is isosceles (definition)