Floor Tessellation: Dealing with Similar Triangles

This image shows the tile pattern on a floor, a collection of similar triangles. This assortment of triangles connects with the AA Postulate, which can be seen by looking at two small triangles. Now, since they coincide perfectly for this tessellation, you can see that the angles are congruent to each other. Since two angles are congruent, then the triangles are similar by the AA postulate.

Now take two small triangles that have connecting vertices that form vertical angles. By the SAS Similarity Theorem, these triangles have one congruent side, congruent angles (vertical), and then another pair of congruent sides. On the other hand, they also demonstrate the SSS Similarity Theorem, because all sides are proportional.

The Triangle Proportionality Theorem is also evident if you look at a larger triangle made of four smaller triangles. A line in the middle that intersects through both sides of the larger triangle is parallel to the base, which divides the sides proportionally.

The Triangle Bisector Theorem can lastly be demonstrated. If you bisect any given triangle with a ray, then it will divide the opposite side into segments proportional to the other two sides.

Overall, many theorems can be applied to this design (~JG2)