Easy Hexagon Explorations

Exploration 1
The dynamic sketch below shows a hexagon ABCDEF with opposite sides parallel (called a parallel-hexagon).
1) What do you notice about its displayed angles?
2) Drag any one of the red vertices to check your observation above. Is it also true if ABCDEF is concave or crossed?
3) Formulate a conjecture and write it in the form "If ..., then ...".

.sketch_canvas { border: medium solid lightgray; display: inline-block; } Hexagon Explorations

Challenge 1
4) Can you explain why (prove) your conjecture in 3) above is true? If stuck, click on the 'Proof Hint' button.
Note: While the result is also true for some types of concave and crossed parallel-hexagons, it is not generally true for all possible types. Can you find such counter-examples?
5) Is the result also true for a octagon or any 2n-gon with opposite sides parallel? Investigate & explain.
6) What about the converse? Formulate it & investigate! If true, explain why it is true. If not, give a counter-example.

Exploration 2
Click on the 'Link to parallelo-hexagon' button to view a parallelo-hexagon ABCDEF, which is a hexagon with opposite sides parallel and equal.
1) Click on the 'Show diagonals' button. What do you notice about the diagonals?
2) Drag any one of the red vertices to check your observation above. Is it also true if ABCDEF is concave or crossed?
3) Formulate a conjecture and write it in the form "If ..., then ...".

Challenge 2
4) Can you explain why (prove) your conjecture in 3) above is true?
5) Is the result also true for a octagon, or any 2n-gon, with opposite sides parallel & equal? Investigate & explain.
6) What about the converse? Formulate it & investigate! If true, explain why it is true. If not, give a counter-example.

Exploration 3
Click on the 'Link to equi-angle-hexagon' button to view an equi-angle-hexagon ABCDEF, which is a hexagon with all its angles equal.
1) Click on the 'Show angles' button. What do you notice about the opposite sides?
2) Drag any one of the red vertices to check your observation above. Is it also true if ABCDEF is concave or crossed?1
3) Formulate a conjecture and write it in the form "If ..., then ...".
Note1: If ABCDEF is dragged into a concave or crossed shape it is necessary to use 'directed angles'.

Challenge 3
4) Can you explain why (prove) your conjecture in 3) above is true? If stuck, click on the 'Proof Hint' button.
5) Is the result also true for a octagon, or any 2n-gon, with all angles equal? Investigate & explain.
6) What about the converse? Formulate it & investigate! If true, explain why it is true. If not, give a counter-example.