The side splitter theorem states that if a line is parallel to a side of a triangle and the line intersects the other two sides, then this line divides those two sides proportionally.

The side splitter theorem is a natural extension of similarity ratio, and it happens any time that a pair of parallel lines intersect a triangle.

No, this example is not accurate. PM is obviously not parallel to OM.

Therefore, the side splitter theorem does not hold and is not true.

$ \frac{LP}{PO} \color{Red}{\ne} \frac{LM}{MN} $

Is the proportion below true?

$ \frac{VW}{WY} = \frac{WX}{YZ} $

No, remember this theorem only applies to the segments that are 'split' or intercepted by the parallel lines.

$ \frac{VW}{WY} \color{Red}{\ne} \frac{WX}{YZ} $

Instead, you could set up the following proportion:

$ \frac{VW}{WY} = \frac{VX}{XZ} $

What if there are more than two parallel lines?

Answer: A corollary of the this theorem is that when three parallel lines intersect two transversals, then the segments intercepted on the transversal are proportional.

Use the corollary to find the value of x in the problem pictured below.

Set up the proportion then solve for x:

Problem 3

Use the corollary to find the value of x in the problem pictured below.

Problem 4

Are the red segments pictured below parallel? (Picture not to scale).

If the red segments are parallel, then they 'split' or divide triangle's sides proportionally. However, when you try to set up the proportion, you will se that it is not true: