Wentworth's Plane and Solid Geometry · Book II, Props. XXXI–XXXIV · Book 3: Book II: The Circle
34. Constructing triangles and parallelograms
Goal: Your student can construct a triangle from a side and two angles, from two sides and an opposite angle (with the ambiguous case), and from three sides, and a parallelogram from two sides and an 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 XXXI–XXXIV (§§ 310–313), with the discussion of when each problem is possible, and Exercises 142–145.
The lesson, step by step
Warm up
Ask: Construct an angle equal to a given angle and explain why it works.
The new idea
To construct a triangle when a side and two angles of the triangle are given.
To construct a triangle when two sides and the angle opposite one of them are given.
To construct a triangle when the three sides of the triangle are given.
To construct a parallelogram when two sides and the included angle are given.
Draw the figure
Prop. XXXII: draw an acute angle A, mark AB = 8 cm on one side, and with radius 6 cm from B draw an arc crossing the other side. It crosses twice: two different triangles fit. This is "the ambiguous case".
Walk through the proof
Each construction rests on a way of proving triangles equal: XXXI on § 139, XXXIII on § 150. Prop. XXXIV uses § 182 (opposite sides equal make a parallelogram).
Go through the discussions together: when is each problem impossible? For XXXIII, when one side is equal to or greater than the sum of the other two.
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
Try Ex. 142–145 on page 119:
Ex. 142. To construct an equilateral triangle, having given the altitude.
Ex. 143. The base and the altitude.
(Ex. 143–145 are isosceles triangles.)
Wrap up
Your student explains, with a sketch, the three cases of Prop. XXXII.
Answers to the book's problems
Ex. 142: at a point of a line erect the altitude; at its top make angles of 30° on each side of it (bisect 60°); the sides meet the line at the base vertices. Ex. 143: erect the altitude at the middle of the base and join its top to the ends of the base.
Worked out for this site from the scan. If a number in your copy differs, trust the book.