Parallel Lines & Proportional Parts
When three or more parallel lines are cut by two transversals, they divide the transversals proportionally. This extends the Side-Splitter beyond a single triangle.
Introduction
The Side-Splitter Theorem (7.2.4) showed that one parallel line inside a triangle creates proportional segments. This lesson generalizes the idea: any set of parallel lines cutting two or more transversals creates proportional segments on every transversal.
Past Knowledge
Side-Splitter Theorem (7.2.4). Parallel lines & transversals (3.1). Proportions (7.1).
Today's Goal
Use parallel lines to find missing lengths on transversals; apply the corollary for equal spacing.
Future Success
Indirect measurement (7.3.1), coordinate proofs, midsegment applications.
Key Concepts
Proportional Parts Theorem
If three (or more) parallel lines intersect two transversals, then they divide the transversals proportionally:
where are on one transversal and are on the other.
Corollary: Equal Spacing
If the parallel lines are equally spaced on one transversal (i.e., ), then they are equally spaced on every transversal ().
Theorem & Proof
Two-Column Proof: Proportional Parts Theorem
Given: Lines cut transversals and at points and respectively
Prove:
Strategy: Construct an auxiliary line to form a triangle, then apply the Side-Splitter Theorem (7.2.4).
| # | Statement | Reason |
|---|---|---|
| 1 | Given | |
| 2 | Draw (connecting point on to point on ) | Two points determine a line |
| 3 | In , line passes through on and some point on , with (i.e., ) | from step 1 |
| 4 | Side-Splitter Theorem in () | |
| 5 | In , means to through ... Similarly, on and on with | from step 1 |
| 6 | Side-Splitter Theorem in () | |
| 7 | Transitive Property: steps 4 and 6 share the ratio |
∎ Parallel lines divide transversals proportionally. The auxiliary diagonal creates the triangles needed for the Side-Splitter.
Worked Examples
Three Parallel Lines
Three parallel lines cut two transversals. On the first transversal, the segments are 6 and 10. On the second, the first segment is 9. Find the second segment.
Cross-multiply: →
Equal Spacing Corollary
Three equally-spaced parallel lines cut a transversal into a segment of 12 and a segment of . A second transversal is cut into segments of 8 and . Find and .
Equally spaced means equal segments on every transversal.
First transversal:
Second transversal:
Four Parallel Lines
Four parallel lines cut transversal into segments 3, 5, 4. They cut into segments with . Find and .
The ratios must match: →
→
Common Pitfalls
Assuming Equal Segments Without Equal Spacing
The segments on the two transversals are proportional, not necessarily equal. They're only equal if the parallels are equally spaced on one of the transversals.
Mismatching Segments Between Transversals
Make sure you're comparing the correct corresponding segments — the first segment on transversal 1 pairs with the first segment on transversal 2 (between the same pair of parallel lines).
Real-Life Applications
City Grid Planning
Streets running parallel to each other cut through diagonal avenues. If the street spacing is equal on one avenue, it's proportionally equal on every diagonal avenue — this is how city planners ensure consistent block sizes.
Sheet Music Staff Lines
The five parallel lines of a musical staff are equally spaced. When a diagonal beam crosses the staff, the equal-spacing corollary guarantees the beam is divided into equal note groupings.
Practice Quiz
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