Tenth Grade · Math · week 22

Proportional relationships in triangles

Apply the Triangle Proportionality (Side-Splitter) Theorem and the Angle Bisector Theorem to solve for unknown segment lengths.

The Triangle Proportionality Theorem (the Side-Splitter Theorem) says that if a line is drawn parallel to one side of a triangle and crosses the other two sides, it divides those two sides into proportional segments. Its converse is also true: if a line divides two sides of a triangle proportionally, that line must be parallel to the third side. This theorem is really similarity in disguise -- the parallel line creates a smaller triangle similar to the original (by AA, using corresponding angles), and proportional sides are just what similar triangles always have.

The Angle Bisector Theorem handles a different cevian: if a ray bisects an angle of a triangle, it divides the OPPOSITE side into two segments that are proportional to the two sides adjacent to the bisected angle. The segment closer to the shorter adjacent side is itself shorter, in that very same ratio -- another proportion the Cross Products Property solves directly.

📋 For the teacher
Teach it (10–15 min):
  • Draw the smaller similar triangle explicitly (a different color) inside the original to show WHY the Triangle Proportionality Theorem is really an AA-similarity fact wearing a new name.
  • For every proportion this week, have them name which theorem justifies the pairing before cross-multiplying -- Triangle Proportionality pairs the two SPLIT sides; Angle Bisector pairs the two ADJACENT full sides with the two split pieces.
  • Use the answer-checking habit from the example (do the two pieces add back up to the whole side?) as a standard last step every time.
  • Sketch the angle bisector case fresh each time, marking the bisected angle with two equal arcs, so "proportional to the ADJACENT sides" has a visual anchor.
Talk about it:
  • Why does a line parallel to one side of a triangle create a smaller triangle similar to the original?
  • How is the Angle Bisector Theorem's proportion different from the Triangle Proportionality Theorem's proportion?
  • Why is checking that your two segment pieces add up to the whole side a good habit, even when the algebra seems to work?

Mastery looks like: They set up and solve proportions correctly using the Triangle Proportionality Theorem and the Angle Bisector Theorem, matching the right sides to the right ratio.

Common stumbles: Pairing the Angle Bisector Theorem's adjacent full sides with the wrong split segments; forgetting to check that computed segment pieces actually sum to the given whole side.

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Records attendance + 15 min of Math for this learner.
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