#### What is the side splitter theorem?

The side splitter theorem states that if a line is parallel to a side of a triangle and intersect 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.

Diagram 1**NOTE:**The side splitter only applies to the intercepted sides. It does not apply to the "bottoms".

#### Is the proportion below true?

**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.

#### Is the proportion below true?

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

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.

**Practice** Problems

To solve this problem, set up the following proportion and solve:

$$
\frac{VW}{WY} = \frac{VX}{XZ}
\\
\frac{7}{14} = \frac{VX}{16}
\\
\frac{16 \cdot 7}{14} = VX
\\
VX = 8
$$

Set up the proportion then solve for x:

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:

$
\frac{10}{12} \color{Red}{\ne} \frac{8}{14}
$

Therefore, the red segments are** not **parallel