Exploring Isosceles Trapezoid (Trapezium) Properties

Exploring Isosceles Trapezoid (Trapezium) Properties (Suggested Grades 4 - 9)

The dynamic geometry activities below are from the "Proof as Explanation" 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: Isosceles Trapezoid Properties Worksheet & Teacher Notes. However, please note that the dynamic activity below does not correspond exactly with the original worksheet activity and has been slightly modified here.

Prerequisites
Knowledge of line (bilateral) symmetry & associated properties. If not already done, it is also strongly recommended, before continuing below, to first complete the following two activities:
Water Supply: Four Towns
Visually Introducing & Classifying an Isosceles Trapezoid

Exploring Isosceles Trapezoid Properties (Suggested Grades 4 - 9)
In this activity, you will explore the properties of an isosceles trapezium (trapezoid), and by dragging, particularly look at its sides, angles, diagonals, slopes, and circle properties.

Explore
A quadrilateral (a closed figure with four straight sides) that looks like the one below is called an isosceles trapezium (trapezoid). In the preceding Visual Exploration of an Isosceles Trapezium you should have seen that an isosceles trapezium (trapezoid) can have different shapes and orientations. In the dynamic sketch below a line AB was constructed, and then two points C and D were reflected across the line to create the isosceles trapezoid CDD'C'.
1) Use the 'Distance', 'Angle', 'Segment', 'Slope', 'Perp. Bisector', and 'Compass' tools on the left (scroll down if necessary) to measure sides, angles, slopes, make constructions, etc.
(Alternatively, click on 'Show Measurement Buttons' to display various buttons for showing measurements and constructions).
2a) What do you notice about its sides, angles, diagonals, symmetry? Does it have a circumcircle or incircle or not? Use dragging to check and confirm your observations.
..b) Also ensure that you drag C across C' (or D across D') to obtain a crossed isosceles trapezoid. What do you notice? Do all the properties in 2a) still hold?

Exploring Isosceles Trapezium (Trapezoid) Properties

Conjecturing & Checking
3a) Click on the 'Link to directed angles' button to navigate to a new sketch showing a crossed isosceles trapezoid C'D'DC with directed angles. What do you notice about its angles in comparison with your earlier observation in 2)? (To learn more about directed angles, go to this activity: Crossed Quadrilateral Angle Sum).
..b)Write down your conjectures in relation to your observations in 1), 2) & 3a).
4) Then check your conjectures by completing this Isosceles Trapezium Properties Quiz which will open in a new browser window. Continue until you get a % score and you are told you've completed the exercise. If you do not get 100%, repeat your dynamic exploration in 1) above.

Logically Explaining
5) Explain (prove) using the symmetry of the isosceles trapezium why your conjectures in 2) to 4) are true.
6) After writing your explanations down, check and compare them with those in this exercise Isosceles Trapezium Properties Explanations or with those in the Teacher Notes.

Notes
a) The intention of the suggested dynamic activities above is not to replace traditional geometric manipulatives such as cardboard, paper, geoboard or plastic representations of various quadrilaterals, and activities like folding & measuring by hand, but rather to supplement and compliment those hands-on concrete activities.
b) This is a suggested activity on Level 2 (Analysis of properties) of the Van Hiele theory of learning geometry. Since definitions according to the Van Hiele theory only start to become meaningful to students at Level 3 (Informal Deduction), note that like in the preceding activity, the concept of 'isosceles trapezoid' is still here introduced without any formal definition; only by name and a dynamic visual representation. For an activity for an isosceles trapezoid on the previous Van Hiele level of learning (Level 1), go to: Visually Introducing & Classifying an Isosceles Trapezoid.
c) For a transitional activity towards the next Van Hiele level of learning (Level 3a: Systematizing/Defining a rhombus), go to: Free construction of an isosceles trapezoid (not yet converted to WSP).
d) Ideally, in the activity at the top, if time allows, learners & students should measure sides, angles, etc. themselves. Alternatively, click on the provided button to display measurements & constructions.
e) For an activity on the next Van Hiele level of learning (Level 3b: Systematizing/Defining an isosceles trapezoid), go to: Systematizing Isosceles Trapezoid Properties - Defining an Isosceles Trapezoid (Rethinking Proof activity).

Dual of Isosceles Trapezoid
As discussed in De Villiers (1996) & (2009), the Kite is the side-angle dual of the Isosceles Trapezoid.

References
De Villiers, M. (1994, 1996, 2009). Some Adventures in Euclidean Geometry (free to download). Lulu Press, Dynamic Mathematics Learning.
De Villiers, M. (1996. An Interesting (Side-Angle) Duality in Geometry. Proceedings of 2nd AMESA Congress, Peninsula Technikon, 1-5 July 1996, 345-350.
De Villiers, M. (1999, 2003, 2012). Rethinking Proof with Geometer's Sketchpad (free to download). Key Curriculum Press.
De Villiers, M. (2011). Simply Symmetric. Learning and Teaching Mathematics, No. 11, pp. 22-26.
De Villiers, M. (2020). The Value of using Signed Quantities in Geometry. Learning and Teaching Mathematics, Dec, no. 29, pp. 30-34.

Other Rethinking Proof Activities
Other Rethinking Proof Activities

External Links
Isosceles trapezoid (Wikipedia)
Equidiagonal quadrilateral (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.)

***************

Free Download of Geometer's Sketchpad & Associated Learning/Instructional Modules on Various Topics

***************

Back to "Dynamic Geometry Sketches"

Back to "Student Explorations"

Created by Michael de Villiers, 3 October 2011 with JavaSketchpad; converted to WebSketchpad, 29 April 2026.