Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity.



We will now use the similarity criteria to establish new and important relationships found in similar triangles such as the 'side splitting theorem' and the angle bisector theorem. We will also prove the Pythagorean Theorem using similar triangles and what eventually comes to be known as the geometric mean relationships.






(1) The student will prove (the side splitting theorem) that a line parallel to one side of a triangle divides the other two proportionally.


(2) The student will prove (the angle bisector theorem) that an angle bisector of an angle of a triangle divides the opposite side in two segments that are proportional to the other two sides of the triangle.


(3) The student will prove the Pythagorean Theorem using similarity and the geometric means.





Being able to establish similarity in geometric shapes allows us to determine angles and establish ratios for corresponding sides. Similar triangles help us to establish how parallel lines partition segments proportionally. These early theorems become very useful in establishing even more concepts as the year develops.




One classic issue appears EVERY YEAR is comparing the proportional pieces cut by the parallel line to the 3rd side that has not been cut into pieces. Students too often relate pieces of sides to whole sides and that causes an error.





The connection to things before this of course is similarity. Any connections to ratios, scale factors, proportion, etc... all connect to the concept of similarity and the non-isometric transformation, dilation.



Parallel lines, proportional relationships, and congruent angles unlock many problems in the future. These new proven theorems will allow students to go at larger problems and have a logical structure to explain why it works.





MY REFLECTIONS (over line l)

The ‘Side Splitting Theorem’, as I have come to know it, where a parallel line to a side of a triangle cuts the other two sides into proportional parts causes students some trouble.  They handle the easy relationships of comparing pieces to pieces but when the ratio of the full length side is used in a ratio they often compare the wrong things.  Do lots of these problems to work out this issue. 

The proof of the angle bisector theorem is a nice proof and a nice use of similarity but it is a pretty useless fact, I can’t think of a time in geometry that it gets used except in its own specific question type.

Finally, the proof of the Pythagorean Theorem using similarity is quite helpful and beautiful. It leads us nicely into the geometric mean relationship.