Side Splitting Theorem of Triangles

In this video, we are going to look at the side splitting theorem of triangles.

Splitting Theorems of Triangles: If a line intersects two sides of a triangle and is parallel to the third side, then it will form two similar triangles.

For example:
Given triangle ABC, if we draw line segment XY across the triangle parallel to line segment AC, then we can see that it is a pair of parallel lines cut by a transversal. From this, we can see that corresponding angles are congruent:
$\angle{A}\cong\angle{BXY}$
$\angle{C}\cong\angle{BYX}$

Notice that the two triangles both share angle B. Along with the two other angles described above, all of the angles in the triangle are congruent, meaning that the two triangles are proportional to each other. As a result
$\frac{\overline{AB}}{\overline{XB}}=\frac{\overline{AC}}{\overline{XY}}=\frac{\overline{BC}}{\overline{BY}}$