Wentworth's Plane and Solid Geometry · Book II, Props. XXVI–XXX · Book 3: Book II: The Circle
33. Copying angles, parallels and dividing a line
Goal: Your student can copy an angle, draw a parallel through a point, divide a line into equal parts, and construct a triangle from two sides and the included angle.
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
Propositions XXVI–XXX (§§ 305–309) and Exercises 138–141.
The lesson, step by step
Warm up
Ask: Bisect an angle while I watch, and say why it works.
The new idea
At a given point in a given straight line, to construct an angle equal to a given angle.
To draw a straight line parallel to a given straight line through a given external point.
To divide a given straight line into a given number of equal parts.
To find the third angle of a triangle when two of the angles are given.
To construct a triangle when two sides and the included angle are given.
Draw the figure
Your student does Prop. XXVI first: an arc on the given angle, the same arc at the new point, then the chord EF carried over with the compasses. Then Prop. XXVIII: divide a 10 cm line into 7 equal parts using a slanting line and parallels.
Walk through the proof
XXVI: the two triangles made by the arcs and chords are equal by three sides, so the angles are equal. (The book asks "Why ?" Your student should answer.)
XXVII: equal exterior-interior angles make the lines parallel (Book I, Prop. XV).
XXVIII: parallels cutting equal parts on one transversal cut equal parts on every transversal (§ 187).
XXIX: the three angles at a point on a line add to two right angles, as do the angles of a triangle.
XXX: two sides and the included angle fix the triangle (§ 143).
Your student proves it back
Close the book. Your student redraws the figure from memory, letters it the same way, and does each construction and explains why it works: 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. 138. To construct an equilateral triangle, having given the perimeter.
Ex. 139. To divide a line into four equal parts by two different methods.
Wrap up
Ask: Why does the line-dividing trick work even if the slanting line is at an odd angle?
Answers to the book's problems
Ex. 138: divide the perimeter into three equal parts (Prop. XXVIII) and build the triangle on one part. Ex. 139: bisect the line and bisect each half; or use Prop. XXVIII with four parts. Ex. 140: the book's hint: draw through the point a perpendicular to the bisector of the angle.
Worked out for this site from the scan. If a number in your copy differs, trust the book.