What is the minimum number of elements we need to show that two shapes must be congruent? The simplest shape we could start with is a triangle. How many elements ( angles , sides) would you need to create a triangle congruent to triangle ABC below? A: Write your hypothesis and explain your reasoning: B: Collect data (i)Try the instructions below and make sure you understand how to work with this applet Select a number of elements of the given triangle ABC (any combination of sides and angles) by clicking on them. Using these elements try to construct a triangle congruent to the given triangle ABC. The segments can be moved by dragging the solid point; rotated by dragging the empty point The angles can be moved by dragging the thicker side; rotated by dragging the empty point or the vertex; the sides can be extended by dragging the endpoints. If your construction has sides parallel to the corresponding sides of the given triangle, you can check your construction by dragging triangle ABC over the constructed triangle. Any element that is deleted cannot be reselected and you will need to refresh and begin a new triangle. (ii) Now plan your data collection. You will need to make choices about the following : What is the simplest data to collect ( remember start simple and then move to complicated) How will you organise your data collect ( tables, screenshots, files) How much data do you need to collect? How will you know that it isn;t possible to make a different triangle with your elements that is not conguent to the original triangle? Write a plan! (iii) Check your plan with a partner and modify it if necessary. (iv) Collect your data!

Conclusion What is your conclusion? Remember to include comments on the accuracy and reliability of your data collection. Also consider the following questions: Are there any other combinations of elements that would work? Have you collected data for all possible elements? What is the minimum number of elements you need ? How do you know? Extension: Investigate the minimum number of elements for other shapes , using a similar process