Systematizing Isosceles Trapezoid Properties - Defining an Isosceles Trapezoid

Systematizing Isosceles Trapezoid Properties - Defining an Isosceles Trapezoid
(Suggested Grades 8 - 10)

"... the describing definition ... outlines a known object by singling out a few characteristic properties." - Hans Freudenthal (1973:458) in Mathematics as an Educational Task.

"Let us remember that mathematics begins with questions, and only ends in axioms (definitions)." - L.W.H. Hull (1969:30) in Mathematics Teaching, 43.

The geometry learning activity below is from the "Proof as Systematization" section of my book Rethinking Proof (free to download).

Worksheet & Teacher Notes
Open (and/or download) a guided worksheet and teacher notes to use together with the dynamic sketch below at: Systematizing Isosceles Trapezoid Properties - Defining an Isosceles Trapezoid Worksheet & Teacher Notes.

Prerequisites
Knowledge of the properties of an isosceles trapezoid - specifically it is expected that you have already completed the following isosceles trapezoid activities:
Visually Introducing & Classifying an Isosceles Trapezoid (Suggested Grades 1 - 7)
Exploring Isosceles Trapezoid (Trapezium) Properties (Suggested Grades 4 - 9)
Free Construction of a Dynamic Isosceles Trapezoid (Suggested Grades 6-10)

Short Description
In the preceding online activity Exploring Isosceles Trapezoid Properties, you should have discovered that an isosceles trapezoid (trapezium) has (among others) the following important properties:

It has two (distinct pairs of equal adjacent angles
It has at least one pair of equal opposite sides
It has at least one axis of symmetry that bisects a pair of opposite sides
Its diagonals are equal
It has at least one pair of parallel sides
It is cyclic (its vertices lie on a circle)

How would you describe what an isosceles trapezoid is, as shortly as possible, over the telephone, to someone who is not yet acquainted with an isosceles trapezoid?

1. Which of the following descriptions (definitions) do you think you would be able to use? Circle these descriptions (definitions).
'.....a. An isosceles trapezoid is any quadrilateral with equal diagonals.
'.....b. An isosceles trapezoid is any quadrilateral with at least one axis of symmetry through a pair of opposite sides and one pair of adjacent angles equal.
'.....c. An isosceles trapezoid is any quadrilateral with at least one pair of sides parallel and at least one pair of opposite sides equal.
'.....d. An isosceles trapezoid is any cyclic quadrilateral with at least one pair of opposite sides parallel.

One way of testing a description (a definition) is to construct a figure complying to the description (definition) to see if it really gives the desired figure. You can test these descriptions (or you own) by using the Construction Tools given in this activity: Free Construction of a Dynamic Isosceles Trapezoid. Otherwise, follow the guided constructions below.

Check by Construction
2. In the dynamic sketch below you are provided with 5 different possible constructions. Press each button step by step on each of the five pages to construct the figures. When each construction is finished, match each page with a descriptions a - d above or in the table given in the Defining an Isosceles Trapezoid Worksheet.
3. Drag each figure & use measurements in the Toolbar on the left (scroll down if necessary) to see if it always remains an isosceles trapezoid. (Note: Since an isosceles trapezoid can be dragged into the shape of a rectangle or a square, we regard a rectangle or a square as special isosceles trapezoids.) In the table given in the accompanying Defining an Isosceles Trapezoid Worksheet, cross out the names of any pages that construct quadrilaterals that are not always isosceles trapezoid.
4. Note that one of these 5 constructions improves on one of the 4 descriptions: Which construction improves on one of the descriptions a–d? Write down the improved description.
5. One of the descriptions a–d has more information than was actually used in the construction. Which one is it? Rewrite that description here, using only the properties used in the construction.

Checking Descriptions/Definitions by Construction

Challenge
Using only logical deduction, can you prove that all the other properties of an isosceles trapezoid listed at the beginning and not included in the correct descriptions (definitions) in Questions 3–5 can be derived from them? Start by using one of these descriptions as your given assumption, then prove as theorems that an isosceles trapezoid has each of the other remaining properties listed at the beginning. Apart from using your description as an assumption in these proofs, you can use any new theorems that you prove in the subsequent proofs of the other properties.

Proving Properties from Chosen Definition
A common, but false misconception is that there is only (correct) definition for each defined object in mathematics; in fact, as we have seen above from various possible constructions, that several different (correct) definitions may exist. Another misconception is that mathematics always starts with definitions — indeed, no definitions of mathematical objects are present a priori in nature (or in some 'ideal' world). In most cases, mathematics starts with problems and only ends in definitions and deductive systems.

Here are three examples of different correct, economical definitions for an isosceles trapezoid, which you may have discovered in the preceding section of this activity:
'.....A. An isosceles trapezoid is any quadrilateral with at least one pair of parallel sides and equal diagonals.
'.....B. An isosceles trapezoid is any quadrilateral with at least one axis of symmetry through a pair of opposite sides.
'.....C. An isosceles trapezoid is any quadrilateral with at least one pair of opposite sides parallel.
Other definitions are possible. Which is best? That is a matter of opinion. But, most importantly, whichever you choose as your (unproven) definition, the properties in the other definitions can be proven as theorems. And all the other properties of an isosceles trapezoid given at the beginning of this activity can be proven starting from any one of the three definitions.
'.....To illustrate this idea we'll start with Definition C above as it is probably least familiar to you as a definition, and answer questions that form the proofs of the other properties of an isosceles trapezoid. We shall restrict ourselves here to the convex cyclic case, but similar proofs can be constructed for the crossed cyclic case.

6. Work through the guided proofs for Definition C as given in the accompanying Defining an Isosceles Trapezoid Worksheet.

Systematize More
7. Now choose either of the Definitions A or B for an isosceles trapezoid and show, as in the guided example 6, how the other properties listed at the beginning and not included in your definition can be proved as theorems.

Explore Further Definitions
8. Investigate, using the tools in the above sketch, or by paper, pencil & compass constructions, whether each of the following is a correct definition of an isosceles trapezoid. If so, provide a complete deductive systematization of the properties of an isosceles trapezoid, similar to those above.
'.....i. An isosceles trapezoid is any cyclic quadrilateral with at least one pair of opposite sides equal.
'.....ii. An isosceles trapezoid is any cyclic quadrilateral with at least one pair of adjacent angles equal.
'.....iii. An isosceles trapezoid is any quadrilateral with at least one opposite sides equal and at least one pair of adjacent angles equal.
'.....iv. An isosceles trapezoid is any quadrilateral with at least one of parallel sides and at least one pair of adjacent angles equal.
'.....v. An isosceles trapezoid is any quadrilateral with with two (distinct) pairs of adjacent angles equal.
'.....vi. An isosceles trapezoid is any quadrilateral with equal diagonals and at least one pair of opposite sides equal.
Note: Check your answers in the Isosceles Trapezoid Teacher Notes.

Class Discussion
A definition can be seen as an agreement among interested parties about what a specific object is. Although you have now seen it is possible to define an isosceles trapezoid in many different ways, it can be very confusing if everyone is using a different definition. It is therefore now necessary to choose a common definition that will be acceptable for the whole class.

9. Have a class discussion to decide which definition of an isosceles trapezoid is most convenient for you.

............

"... the algorithmically constructive and creative definition ... models new objects out of familiar ones." - Hans Freudenthal (1973:458) in Mathematics as an Educational Task.

isosceles trapezoid specializing generalizing

Defining and Investigating New Concepts
New mathematical objects are often defined by modifying or extending the definitions or properties of known objects in mathematics. This is known as constructive (a priori) defining. The following are some possible examples in relation to the concept of an isosceles trapezoid.
'.....10a. Investigate whether a trilateral trapezoid, that is, an isosceles trapezoid with at least three equal sides as shown in the 1st figure above, has any interesting properties in addition to those of an isosceles trapezoid. (Hint: Specifically look at how the diagonals divide the angles.)
'.....10b. Prove your observations in 10a.
'.....10c. Prove that a cyclic quadrilateral with at least three equal sides has at least one pair of opposite sides parallel. (In other words, that it is a trilateral trapezoid.)
'.....11a. Investigate the properties of an isosceles hexagon, that is, a hexagon with at least one axis of symmetry through a pair of opposite sides as shown in the 2nd Figure above.
'.....11b. Prove your observations in 11a. Can you generalize further?
Note: Check your answers in the Isosceles Trapezoid Teacher Notes.

Pedagogical Notes
In earlier activities, we have seen that proof can have the following functions (roles) in mathematics:
'.....- explanation (providing insight into why something is true)
'.....- discovery (assisting in the discovery or invention of new results)
'.....- verification (checking the validity or truth of a statement)
'.....- challenge (challenging the ingenuity of the problem solver)

In this activity, however, proof was used in a completely different way. Here we focused on organizing the properties of an isosceles trapezoid into a logical system consisting of a definition (an unproved statement) and theorems (proved statements). The purpose was not to verify that these properties were true (we already know from earlier investigations that they are true), but to investigate and compare different possible logical organizations.
'.....In this sense, proof is now being used as a means of systematization (that is, as a tool to logically organize previously unrelated properties or statements into a deductive system).

References
De Villiers, M. (1994, 1996, 2009). Defining and Classifying (Chapter 2), pp. 13-28, from Some Adventures in Euclidean Geometry. (Free to download). Morrisville, NC: Lulu Publishers, Dynamic Mathematics Learning.
De Villiers, M. (1998). To Teach Definitions in Geometry or Teach to Define? In A. Olivier & K. Newstead (Eds), Proceedings of the Twenty-second International Conference for the Psychology of Mathematics Education: Vol. 2. (pp. 248-255). University of Stellenbosch: Stellenbosch, 12-17 July 1998.
De Villiers, M. (2004). Using dynamic geometry to expand mathematics teachers’ understanding of proof. Int. J. Math. Educ. Sci. Technol, Vol. 35, No. 5, 703–724.
De Villiers, M.; Govender, R. & Patterson, N. (2009). Defining in Geometry. In T. V. Craine & R. Rubenstein (Eds.). (2009). Understanding Geometry for a Changing World: NCTM's 71st Yearbook. Reston, VA: NCTM.
De Villiers, M. (1999, 2003, 2012). Rethinking Proof with Geometer's Sketchpad (free to download). Key Curriculum Press.
Freudenthal, H. (1973:458) in Mathematics as an Educational Task. Dordrecht: D. Reidel.
Hull, L.W.H. (1969). The superstition of educated men, Mathematics Teaching, 43, 26 - 31.
Human, P.G. & Nel, J.H. et al. (1987). Alternative Instructional Strategies for Geometry Education: A Theoretical and Empirical Study. Final report of the University of Stellenbosch Experiment in Mathematics Education (USEME)-project: 1977-78. (Only the theoretical part of the report is included here.)

Other Rethinking Proof Activities
Other Rethinking Proof Activities

External Links
Van Hiele model (Wikipedia)
Isosceles trapezoid (Wikipedia)
Aimssec Lesson Activities (African Institute for Mathematical Sciences Schools Enrichment Centre)
UCT Mathematics Competition Training Material
SA Mathematics Olympiad Questions and worked solutions for past South African Mathematics Olympiad papers can be found at this link.
(Note, however, that prospective users will need to register and log in to be able to view past papers and solutions.)

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Michael de Villiers, created with WebSketchpad, 15 May 2026.

Systematizing Isosceles Trapezoid Properties - Defining an Isosceles Trapezoid (Suggested Grades 8 - 10)

Systematizing Isosceles Trapezoid Properties - Defining an Isosceles Trapezoid
(Suggested Grades 8 - 10)