If several straight lines are cut by two transversal lines, the ratio of any two segments of one of these transversals is equal to the ratio of the corresponding segments of the other transversal. The following Descartes window shows how three parallel straight lines are cut by two secants r and s. The window indicates the length of the segments in these two transversals at any time, and that the ratio of the segments does not change.

2.- Move the parallel line in the middle by dragging the red point with the mouse and notice how the ratio values change.

3.- Copy this example into your exercise book, with the same measurements, and measure the segments to see if the ratio values are equal or not.

4.- Move the parallel line in the middle until segment AB is equal to BC. Then, see if the segment A'B' is equal to B'C'. Move the lines r and s and see if the segments remain equal to each other.

If points A and A' had met in the window above they would have formed a triangle with points C and C'. Thales' theorem would have still been satisfied which allows us to conclude that: Any line parallel to one of the sides of a triangle, which cuts the other two sides, produces segments which are in proportion to each other. In the Descartes' window there is a triangle ABC and a parallel line to side BC which passes through points D and E producing segments which are in proportion to each other, as they are in the same ratio.

5.- Click on the Init button and draw an identical triangle to the one in the window into your exercise book. Draw the line parallel to side BC and check the measurements and ratio values. Verify that the ratio values stay the same when points B and C are moved horizontally, but that they do change when the we move the straight line.

6.- Move the straight line onto vertex A and note that the segments are still in proportion to each other. This is also the case when the line is dragged below points B and C.

7.- A triangle has sides AB=10 cm, AC=12 cm and BC=8 cm. A line parallel to side BC is drawn 4 cm away from vertex A along side AB, which cuts the triangle at points D and E. Work out the lengths of AD, AE and DE.

Two triangles are similar when their angles are equal and their sides are in the same ratio. In other words, if triangles ABC and A'B'C' are similar then A=A', B=B' and C=C', and the ratio values A'B'/AB=B'C'/BC=C'A'/CA=r, which is known as the ratio of similitude. 

9.- If the sides of the green triangles were 3, 4 and 5 what would be the length of the sides of the blue one?

10.- Alter the ratio until its value is 1 and you will see that the triangles are identical in size and shape.

11.- Reduce the ratio to 0.5 and compare the triangles. How long are the sides of the green triangle if the sides of the blue triangle are 3, 5 and 7?

12.- Repeat the process for ratios of 1.5, 0.25 and 3. Change the scale to 16 in the last example so that you can see both triangles.

13.- Are two equal triangles similar? What about two equilateral triangles?

14.- Are two triangles similar if their angles are equal? Are two triangles similar if their sides are in the same ratio?

Two polygons are similar if their angles are equal and their sides are in the same ratio. In other words, if polygons ABCDE and A'B'C'D'E' are similar then angles: A=A', B=B', C=C', D=D' and E=E', and the ratio values A'B'/AB=B'C'/BC=C'D'/CD=D'E'/DE=E'A'/EA=r, which is known as the ratio of similitude. 

16.- Change the ratio to 1 and note that the pentagons are identical in size and shape. Change the ratio to 2 and compare the pentagons. What are the measurements of the green pentagon if the sides of the blue one are 3, 5, 6, 8 and 7?

17.- Repeat the process for ratios of 2,3 and 0.25. Change the scale to 16 in the last example so that you can see both pentagons.

18.- Are two equal pentagons similar? What about two regular pentagons? Are two pentagons similar if their angles are equal? Are two pentagons similar if their sides are in the same ratio?