Wentworth's Plane and Solid Geometry · Book III, §§ 351–358 (Props. XVII–XIX) · Book 4: Book III: Proportion. Similar Polygons

44. Similar polygons and similar triangles

Goal: Your student can define similar polygons and prove the three tests for similar triangles.

⏱ About 40 minutes · 6 steps

Open it at the Internet Archive ↗

You'll need

  • The book, open to these pages
  • A ruler (straightedge) and a pair of compasses
  • Sharp pencil, eraser and plain paper (or a geometry notebook)
  • A protractor

What's in the book

The definition of similar polygons (§§ 351–353) and Propositions XVII–XIX (§§ 354–358) with corollaries.

The lesson, step by step

  1. Warm up

    Ask: State the theorem on the bisector of an angle of a triangle.

  2. The new idea

    Read §§ 351–353. Similar polygons need both equal angles and proportional sides; the book shows figures with only one of the two.

    Two mutually equiangular triangles are similar.

    If two triangles have an angle of the one equal to an angle of the other, and the including sides proportional, they are similar.

    If two triangles have their sides respectively proportional, they are similar.

  3. Draw the figure

    Draw triangle ABC. Draw a smaller triangle with the same angles (copy two angles). Then "place" the small one in the corner A of the big one, as the book does: it sits as triangle AEH with EH parallel to BC.

  4. Walk through the proof

    XVII: in the corner, angle AEH = angle B, so EH is parallel to BC, and by § 343 the sides are proportional.

    XVIII: placed in the corner, the proportional sides make EH parallel to BC (Prop. XIV), so the angles are equal.

    XIX: take AE = A'B' and AH = A'C'. Triangle AEH is similar to ABC (Prop. XVIII), so EH = B'C', and AEH equals A'B'C' (three sides).

  5. Your student proves it back

    Close the book. Your student redraws the figure from memory, letters it the same way, and gives the proof aloud: what is given, what is to be proved, then each step with its reason ("because the radii are equal", "by § 143"). Point to each line or angle as it is named.

    If a step is missing, ask "How do we know that?" rather than supplying it.

  6. Wrap up

    Ask: Are two right triangles with a 30° angle similar? Why? (Yes, § 356.)

@teddyrookbook GitHub