Summary
Highlights
The proportionality theorem applies to triangles with two parallel lines. If two lines within a triangle are parallel, specific ratios of the side lengths will be equal. For example, AB/BC = AD/DE. The key is that the ratio, not the absolute length, remains the same.
You can form various combinations of ratios. For instance, AB over the whole length AC will be equal to AD over the whole length AE. The order of the corresponding segments must be maintained on both sides of the equation. Whether you write BC or CB for a segment does not affect the correctness.
To solve for an unknown, such as 'x', when given a triangle with parallel lines (e.g., DE parallel to BC), you set up the proportional ratios. For example, AB/BC = AD/DE. The reasoning for this step in a test can be 'prop theorem' along with stating which lines are parallel (e.g., DE || BC), or 'line parallel to side of triangle' which implies the same proportional relationship.
Substitute the given values into the established proportion. For example, (x+2)/3 = (x+8)/7. Then, cross-multiply to solve for x: 7(x+2) = 3(x+8). Expand and simplify the equation: 7x + 14 = 3x + 24. Finally, solve for x, which in this case is x = 2.5.