Quadrilaterals

A quadrilateral is any four-sided closed plane figure. Some quadrilaterals have particular names and properties, and those quadrilaterals are generally known as special quadrilaterals.

Figure 1

In Figure 1 there are two quadrilaterals which show the justification for a property called convex. This property means that the plane figure has no “indentations.” The justification involves choosing any two interior points of the polygon and connecting those points with a segment. If the segment is contained in the interior of the polygon (as in the figure on the right), the polygon is convex. If the segment is not fully contained in the polygon (as in the figure on the left), the polygon is nonconvex. Some texts will use the term concave for the figure on the left.

Figure 2

In Figure 2 let’s focus on vertex A. Vertices B and D are said to be consecutive vertices with A and vertex C is said to be nonconsecutive with A. What is a diagonal of a plane figure? A diagonal is an interior segment that joins two nonconsecutive vertices of the polygon. How many diagonals does a quadrilateral have?

In an earlier investigation, we found that the sum of the interior angles of a triangle is 180º. How can we use this information to find the sum of the interior angles of a quadrilateral?

Figure 3

First, draw a single diagonal of the quadrilateral so the triangles formed are not overlapping, as shown in Figure 3. What can you say about the triangles and the sum of the interior angles of the quadrilateral? If you said that the angles total 360º, you’d be CORRECT!

Special Quadrilaterals

The special quadrilaterals are the parallelogram, rectangle, rhombus, square, trapezoid, and kite. Each of these has the properties attributed to the quadrilateral, i.e. a four-sided, closed polygon whose interior angles sum to 360º. Each of these special quadrilaterals will also have 2 diagonals.

Parallelogram

· 2 pair of opposite sides parallel and equal in length

· 2 pair of opposite angles equal in measure

· consecutive angles supplementary

· diagonals bisect each other (Bisect means cuts in half.)

Rectangle

· a parallelogram

· 4 right angles

· diagonals of equal length

Rhombus

· a parallelogram

· 4 equal sides

· diagonals perpendicular

Square

· a parallelogram

· 4 right angles

· 4 equal sides

· diagonals of equal length and perpendicular

Trapezoid

· one pair of opposite sides parallel

· When will the diagonals be equal length?

Kite

· 2 pair of congruent consecutive sides on either side of the line of

symmetry

· diagonals perpendicular

Note: On the special quadrilaterals that say “a parallelogram,” this property tells you that all the properties of the parallelogram hold for that given quadrilateral.