Parallel Lines and Proportional Parts
Overview
Parallel lines cut any transversals through them proportionally, and the same idea works inside a triangle: a line parallel to one side cuts the other two sides into proportional segments. This lesson applies proportions to those intercepted segments, revisits the midsegment as the halfway case, and adds one more proportional split — an angle bisector divides the opposite side into segments proportional to the two sides of the triangle.
Vocabulary
- Proportional Segments
- Segments whose lengths form a proportion; parallel lines cut their transversals into proportional segments, so the segments between the parallel lines on one transversal correspond to the segments on the other.
- Transversal
- A line that intersects two or more other lines at different points, all in the same plane; parallel lines cut their transversals into proportional segments.
- Midsegment of a Triangle
- A segment that joins the midpoints of two sides of a triangle; it is parallel to the third side and its length equals one half the measurement of the third side.
- Triangle Angle Bisector Theorem
- In a triangle, an angle bisector cuts the opposite side into segments that are proportional to the two sides of the triangle: AB/AC = BD/DC for a bisector from A meeting BC at D.
Main Ideas
- Parallel lines cut their transversals proportionally: set up the proportion by matching segments that lie between the same pair of parallel lines.
- A line through a triangle parallel to one side divides the other two sides proportionally (the same proportion read from the vertex: part-to-part or part-to-whole, as long as both sides use the same reading).
- The midsegment is the halfway case: it connects the midpoints, is parallel to the third side, and equals half the third side.
- An angle bisector of a triangle divides the opposite side proportionally to the adjacent sides.
- Every problem in this lesson is one proportion: match the corresponding segments, cross-multiply, and check by substituting back.
Lesson Notes
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Lesson Video
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Practice Problems
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Download the practice worksheet for this lesson, work the problems, then check your work with the solutions.