Triangle Proportionality Theorem
Also called the Side-Splitter Theorem: a line parallel to one side of a triangle divides the other two sides proportionally.
Introduction
When a line is drawn inside a triangle parallel to one of its sides, it creates a smaller triangle that is similar to the original (by AA). This gives us a powerful proportion: the parallel line splits the two sides it crosses into segments with the same ratio.
Past Knowledge
AA Similarity (7.2.1). Parallel lines & transversals (3.1). Proportions (7.1).
Today's Goal
Use the Side-Splitter Theorem and its converse to find missing lengths and prove lines parallel.
Future Success
Parallel lines & proportional parts (7.2.5), midsegment revisited, coordinate proofs.
Key Concepts
Triangle Proportionality Theorem (Side-Splitter)
If in (with on and on ), then:
Converse
If , then .
Alternative Form
You can also write: (part to whole).
Theorem & Proof
Two-Column Proof: Triangle Proportionality Theorem
Given: In , with on and on
Prove:
Strategy: Use AA to show , then derive the side-segment proportion.
| # | Statement | Reason |
|---|---|---|
| 1 | Given | |
| 2 | and | Corresponding Angles Postulate (, transversals and ) |
| 3 | AA Similarity (step 2) | |
| 4 | Corresponding sides of similar triangles are proportional | |
| 5 | Reciprocal Property of Proportions (7.1.2) | |
| 6 | Subtract-the-Denominator Property (applied to step 5 → subtract 1 from both sides) | |
| 7 | , so | Segment Addition: and ; then take reciprocals |
∎ A line parallel to one side of a triangle divides the other two sides proportionally.
Worked Examples
Finding a Missing Segment
In , with . Find .
By the Side-Splitter Theorem:
Cross-multiply: →
Is the Line Parallel?
. Is ?
The ratios are equal → by the converse, .
Yes,
Algebraic Unknown
with . Find .
Cross-multiply:
→
(so )
Common Pitfalls
Setting Up the Wrong Proportion
The theorem says (segment-to-segment). Don't mix segments from different sides in the wrong places.
Forgetting the Parallel Requirement
The Side-Splitter Theorem only works if the line is parallel to the base. Without that condition, the proportionality relationship doesn't hold.
Real-Life Applications
Land Surveying — Property Lines
When a road cuts through triangular land parcels parallel to one boundary, the Side-Splitter determines how the two side boundaries are divided — critical for fair property division.
Staircase Design
Stair treads are parallel horizontal lines cutting through the diagonal profile. The Side-Splitter ensures equal rises and runs — each “slice” divides the diagonal proportionally.
Practice Quiz
Loading...