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On Quadrilaterals Let Them Explore and Guess Michael Keyton IMSA [email protected] On Quadrilaterals In high school geometry courses, the study of quadrilaterals is limited. Usually, only six (sometimes at many as eight) quadrilaterals are defined and studied peripherally. The six studied are the trapezoid, isosceles trapezoid, parallelogram, rhombus, rectangle, and square (sometimes kite and cyclic quadrilateral). We will look at many more, discuss definitions, and make suggestions for students to make the study of geometry more mathematical. On many occasions, I have argued and supported inclusive definitions for each of these. Historically, they were all given exclusively, but as time passed all now have inclusive relationships except for the trapezoid and isosceles trapezoid. Typical definitions for the quadrilaterals are: Trapezoid - a quadrilateral with exactly one pair of parallel sides (I will argue to change this definition and supply reasons for why it has not been universally done) Isosceles trapezoid – a trapezoid with congruent non-parallel sides (again suggestions for change) Parallelogram – a quadrilateral with both pairs of opposite sides parallel Rectangle – a parallelogram with a right angle (this has been changed over the years, to be discussed later) Rhombus – a parallelogram with congruent consecutive sides Square – a rectangle with congruent consecutive sides. In Euclid, rectangles and rhombus were not allowed to be parallelograms, squares were not allowed to be rectangles or rhombi. All definitions of quadrilaterals were exclusive. Part I – Inclusive Definitions and Definitions Inclusive definitions are much more powerful mathematically than exclusive ones. Having definitions for the rectangle such as above serves three outstanding mathematical purposes – first, if A is defined to be a subclass of B, then any theorem proved for B becomes automatically true for A; second, this gives more new theorems; and third, a converse does not require an or conclusion. For example, with exclusive definitions such as above, the following theorem causes difficulty. If the diagonals of a quadrilateral separate each other proportionally, then the quadrilateral is a trapezoid or a parallelogram. So why do we still have a few remnants of exclusive definitions. I believe because in high school geometry courses, there are very few theorems for the quadrilaterals. For the trapezoid there is usually only two theorems given: (1) The median of a trapezoid is the arithmetic average of the parallel sides. (2) the area of a trapezoid is one half the product of the height and the sum of the bases. In an earlier paper, I gave a list of theorems about trapezoids. Definitions are arbitrary. They can be changed, but there are characteristics that make some better than others. A good definition should have the following properties: Keyton 2 IMSA (1) It should be define the new object using known terms (previously defined words or undefined words). (2) It should be minimal, requiring as few parts to verify as possible, (3) it must be reversible, (4) it should distinguish the object being defined from other objects (5) it should be inclusive. `When I use a word,' Humpty Dumpty said, in rather a scornful tone, `it means just what I choose it to mean -- neither more nor less.' Lewis Carroll, Through the Looking Glass An example of a poor definition for a rectangle is a quadrilateral with four right angles. This violates the minimal condition, for to verify that a quadrilateral is a rectangle requires four tests, when only three are needed. A better definition is a rectangle is a quadrilateral with three right angles. This allows a theorem – if a quadrilateral has three right angles, then it has four right angles. If a theorem and its converse about a quadrilateral X are both true, then the definition of the quadrilateral can be replaced with the theorem. Thus, we see the following results and converses about parallelograms: (1) a quadrilateral with both pairs of opposite sides parallel (2) a quadrilateral with both pairs of opposite sides congruent (3) a quadrilateral with diagonals that bisect each other (4) a quadrilateral with both pairs of opposite angles congruent (5) a quadrilateral with a pair of opposite sides congruent and parallel (6) a quadrilateral with all a pair of opposite angles that are congruent and supplementary to one of the other angles There are many more that could be stated. Which one is the best definition for a parallelogram? The answer is any. But (1) conveys the linguistic meaning of the word "parallelogram." For the others, should that be thought of as quadrilaterals (Four sided figures) or quadrangles (four angled figures), or just figures. In alignment with these characteristics, I offer the following as definitions of the quadrilaterals. A cyclic quadrilateral is a quadrilateral with vertices on a circle. A trapezoid is a quadrilateral with a pair of parallel sides. A parallelogram is a quadrilateral with two pairs of parallel opposite sides. A rectangle is a quadrilateral with 3 right angles. An isosceles trapezoid is a quadrilateral with two disjoint pairs of consecutive congruent angles. A kite is a quadrilateral with two disjoint pairs of consecutive congruent sides. A rhombus is a quadrilateral with congruent sides. A square is a quadrilateral with congruent sides and a right angle. Keyton 3 IMSA Notice, the complexity of the figure depends upon the number of conditions that it needs to satisfy. A trapezoid and the cyclic quadrilateral need only 1 condition; parallelograms, kites, and isosceles trapezoids need two; rectangles and rhombi need three; and the square needs four. Part II Other quadrilaterals Why are we restricted to just these quadrilaterals? I offer the following as some of the reasons. The square is the basic unit for measuring area. It also has the maximum symmetry. The rectangle is an immediate generalization of the square for area. The parallelogram allows us to transfer angles and distances in the plane. The rhombus transfers distances in two directions. It is the intersection of two congruent circles. The trapezoid is an easy way to measure areas. (not in Euclid) The kite is formed by the intersection of two circles. It also has a line of symmetry. (not in Euclid) The isosceles trapezoid has symmetry. The cyclic quadrilateral transfers angles through space. Are there others? Since Euclid, we have added the cyclic quadrilateral, trapezoid, and kite. Others which appear some in advanced studies are: (a) An orthodiagonalquadrilateral is a quadrilateral with perpendicular diagonals. (b) An isodiagonalquadrilateral is a quadrilateral with congruent diagonals. Number of characteristics: (1) A ___________ is a quadrilateral with a right angle. A ____________ is a quadrilateral in which one diagonal bisects the other. (2) A ____________ is a quadrilateral with perpendicular opposite sides. (2) A ____________ is a quadrilateral with perpendicular and congruent diagonals. The objective is not only to define but also to find theorems about each of the new quadrilaterals. Keyton 4 IMSA III Other investigations of existing quadrilaterals. The investigations in the books are about the angles, the sides, and the diagonals. How about including (a) the intersection point of the diagonals (the diacenter), (b) the medians (segments from midpoints of opposite sides, (c) the intersection point of the medians (medcenter), (d) the segments from midpoints of adjacent sides (???), (e) segments from the vertices to a midpoint of an opposite side, (f) quadrilaterals formed by the intersections of segments in (e), (g) altitudes from vertices to opposite sides (note there are 8), (h) quadrilaterals formed by the intersections of the lines in (g), (i) angle bisectors, (j) quadrilaterals formed by intersections of lines in (i), (k) the centroids, circumcenters, incenters, orthocenters of the 8 triangles formed by the diagonals and the sides of the quadrilateral, (l) the quadrilaterals formed by the intersections of the points in (k), (m) parallels through the vertices to the diagonals, (n) perpendiculars to the diagonals from the vertices of the quadrilateral, In general if we think of a property X and a quadrilateral Z, we can form two theorems: If Z has property X, then the result is ______________ In the article “Students Discovering Geometry using Dynamic Geometry Software” in the book Geometry Turned On (MAA, 1997), I give ideas for about 200 different explorations that students can pursue on quadrilaterals. In the years I have used explorations using software, I have seen more than 1000 new results from students. Some other papers can be found at www.imsa.edu/~keyton in the geometry section. The main results that I have found with and for students is that they begin to appreciate the necessity of proof, the difficulty with creating definitions and stating conjectures, the pleasures and the agony in the discovery methods, seeing how immense geometry is, and demonstrating their creativity beyond what I could ever conceive. Never overestimate what a student knows; never underestimate what he can produce. Michael Keyton Keyton 5 IMSA