## Rotated Lines

Construction & Conjecture - Triangle
1) Lines AB, BC and CA of triangle ABC have been rotated through the same (blue) angle DEF, respectively around B, C and A as shown.
2) What do you notice about triangles ABC and GHI? Check your conjecture by clicking on the 'Show Angle Measurements GHI' button.
3) Check your conjecture further by dragging points A, B or C to change triangle ABC, or F to change the angle of rotation.

Challenge
4) Can you explain why (prove) that triangles ABC and GHI are similar?
5) Formulate a converse and investigate. If true, explain why (prove) or if false, give a counter-example.

#### .sketch_canvas { border: medium solid lightgray; display: inline-block; } Rotated Lines

1) Investigate what happens if the same construction as for a triangle is carried out with a quadrilateral.
(Think about it before clicking on the 'Link to quadrilateral' button on the bottom right of the sketch).
2) What do you notice about quadrilaterals ABCD and KLMN? Are they still similar?
3) Click on the 'Show Ratio measurements' button to check your observation above. What do you notice about the ratios of the corresponding sides? What does this tell you?
4) Click on the 'Show Angle Measurements KLMN' to show the angles of KLMN. What do you notice? Formulate a conjecture.
5) Firther check your conjecture by dragging points A, B, C or D to change quadrilateral ABCD, or G to change the angle of rotation.

Challenge
6) Can you explain why (prove that) your observations above are true?
7) Formulate a converse and investigate. If true, explain why (prove) or if false, give a counter-example.

Generalize Further
Can you generalize further to pentagons, hexagons, etc.? What happens if the lines AB, BC and CA are rotated through the same angle around points other than the vertices?

Created 17 Nov 2007 by Michael de Villiers; updated to WebSketchpad, 27 August 2022.