Two polygons are similar if all their corresponding angles are congruent, or all their corresponding sides are proportional. For instance, we can say that triangle is similar to triangle if one of the following conditions hold:
Condition 1: and and
Condition 2: and and .
In this post, we are going to use the concept of similarity to prove the Side Splitter Theorem. The Side Splitter theorem states that if is any triangle, and is drawn parallel to , then .
The Side Splitting Theorem
If is any triangle, and is drawn parallel to , then .
Draw parallel to .
and by the Parallel Line Postulate. Note that is parallel to and acts a transversal.
Now, by AA Similarity, triangle is similar to triangle .
By definition of similarity,
But notice that is a parallelogram (why?), so it follows that .
Therefore, which is what we want to prove.