Wentworth's Plane and Solid Geometry · Book III, Props. XX–XXIII · Book 4: Book III: Proportion. Similar Polygons
45. More similar triangles: parallel sides, altitudes, transversals
Goal: Your student can prove that triangles with sides parallel or perpendicular are similar, that altitudes are in the ratio of the sides, and the theorems on transversals through a point.
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)
What's in the book
Propositions XX–XXIII (§§ 359–363) and Exercise 251.
The lesson, step by step
Warm up
Ask: Give the three tests for similar triangles.
The new idea
Two triangles which have their sides respectively parallel, or respectively perpendicular, are similar.
The homologous altitudes of two similar triangles have the same ratio as any two homologous sides.
If two parallels are cut by three or more transversals that pass through the same point, the corresponding segments are proportional.
Prop. XXIII (§ 363) is the converse: lines cutting proportional segments on two parallels pass through a common point.
Draw the figure
Draw a point O and four lines spreading out from it. Cut them with two parallel lines, and label the crossing points A, B, C, D and A', B', C', D'.
Walk through the proof
XX: with parallel sides, the angles are equal by the parallel-line theorems; with perpendicular sides, by supplements in a quadrilateral with two right angles.
XXI: the altitudes cut off right triangles with an equal acute angle, so they are similar (§ 356).
XXII: the pairs of triangles with vertex O are similar, so each segment ratio equals OB : OB' and so on.
XXIII: extend two lines to meet at O, join O to the third point, and show the third line goes through O.
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.
Exercises
Ex. 251. The base and altitude of a triangle are 7 feet 6 inches and 5 feet 6 inches, respectively. If the homologous base of a similar triangle is 5 feet 6 inches, find its homologous altitude.
Wrap up
Read the note after Prop. XXII on writing a chain of ratios in condensed form.