refresher

def
A quadrilateral is a four-sided polygon.

All quadrilaterals are coplanar and have two diagonals.

Quadrilaterals may be drawn in a "convex" form or a "concave" form. The quadrilaterals we will be using will be convex, where the diagonals lie within the figure. A concave quadrilateral contains an angle greater than 180º, and at least one of the diagonals lies partially, or entirely, outside of the figure (as shown below). The concave figure "caves in" upon itself.

quadcave1

Theorem
The sum of the measures of the four angles of any quadrilateral equals 360º.
In the diagram at the right,
m∠A + m∠B + m∠C + m∠D = 360º.
Every quadrilateral can be decomposed into two triangles, each of whose angles' measures sum to 180º.
m∠A + m∠1 + m∠ 3 = 180º
m∠C + m∠2 + m∠4 = 180º
m∠A+m∠1+m∠3+m∠C+m∠2+m∠4 = 360º
quadanglesquadangles2

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There are several "types" of quadrilaterals determined by their characteristics, or properties.
We will be examining three types of quadrilaterals: parallelograms, trapezoids and kites.

Parallelograms:

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A parallelogram is a quadrilateral with both pairs of opposite sides parallel.

There are three additional quadrilaterals that possess the properties of a parallelogram.
pararefresh1
Definition: A parallelogram has two pairs of parallel sides.
• The opposite sides of a parallelogram are congruent.
• A parallelogram has no reflectional symmetry.
• A parallelogram has rotational symmetry of 180º (Order 2).
rectrefresh
Definition: A rectangle is a parallelogram with 4 right angles.
• A rectangle has two pairs of parallel sides.
• A rectangle has opposite sides congruent.
• A rectangle also has four right angles.
• A rectangle has two axes of reflectional symmetry.
• A rectangle has rotational symmetry of 180º (Order 2).
rhombusrefresh
Definition: A rhombus is a parallelogram with 4 congruent sides.
• A rhombus has two pairs of parallel sides.
• A rhombus has all four sides congruent.
• A rhombus has 2 (diagonals) axes of reflectional symmetry.
• A rhombus has rotational symmetry of 180º (Order 2).
squarerefresh
Definition: A square is a parallelogram with 4 congruent sides and 4 right angles.
• A square has two pairs of parallel sides.
• A square has all four sides congruent.
• A square has four right angles.
• A square has four axes of reflectional symmetry.
• A square has rotational symmetry of 90º (Order 4).
Read more about symmetries at Symmetry in Geometry.

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Trapezoids:

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A trapezoid is a quadrilateral with at least one pair of parallel sides.

There are two basic types of trapezoids: regular trapezoids and isosceles trapezoids.
trapref
Definition: A trapezoid has at least 1 pair of parallel sides.
• A trapezoid has no axes of reflectional symmetry.
• A trapezoid has no rotational symmetry (Order 1).
Note: Having no rotational symmetry means that the figure must be rotated a full 360º to again appear in it original position. This may be referred to as the identity symmetry.

isostrapref
Definition: An isosceles trapezoid is a trapezoid with congruent base angles.
• An isosceles trapezoid has congruent legs.
• An isosceles trapezoid has at least 1 axis of reflectional symmetry. (If it has 2, it is a rectangle.)
• An isosceles trapezoid (with only 1 pair of parallel sides) has no rotational symmetry.

Note: The definition of an isosceles triangle states that the triangle has two congruent "sides". But the definition of isosceles trapezoid stated above, mentions congruent base "angles", not sides (or legs). Why?
If an isosceles trapezoid is defined to be "a trapezoid with congruent legs", a parallelogram will be an isosceles trapezoid. If this occurs, the other properties that an isosceles trapezoid can possess can no longer hold, since they will not be true for a parallelogram.
If we define an isosceles trapezoid to be a trapezoid with congruent base angles, we can easily prove the sides (legs) to also be congruent, a parallelogram will NOT be an isosceles trapezoid, and all of the commonly known properties of an isosceles trapezoid will still be true.

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Kites:

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A kite is a quadrilateral whose four sides are drawn such that there are two distinct sets of adjacent, congruent sides.

realkite

• A quadrilateral kite resembles a flying kite.

• A kite has two pairs of adjacent sides of equal measure. Notice that these sides are not "opposite" sides.

• The kite at the right has one vertical axis of reflectional symmetry, bd.

• The kite has no rotational symmetry.

kitequad

A kite also resembles a triangle reflected over one of its sides, forming a quadrilateral.
Notice ΔABD being reflected over its side from B to C, forming ΔCBD.

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