Wentworth's Plane and Solid Geometry · Book I, §§ 214–215, Exs. 19–27 · Book 2: Book I: Rectilinear Figures
19. Methods of proving theorems
Goal: Your student understands the synthetic, analytic and indirect methods, and works through the book's model proofs of Exercises 19–27.
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
Sections 214–215 explain the three methods of proof and give advice on auxiliary lines. Exercises 19–27 then show proofs worked out, with "(Why ?)" left for the student to answer.
The lesson, step by step
Warm up
Ask your student to state the theorem about the sum of the angles of a triangle and prove it.
Three methods
Read § 214 together.
The synthetic method is the method employed in most of the theorems already given, and consists in putting together known truths in order to obtain a new truth.
The analytic method works backward from what we want, step by step, to something known. The indirect method proves the contradictory false (as in Prop. XXVII).
Useful habits
Read § 215: how lines and angles are usually proved equal, and the two suggestions:
1. Draw as accurate figures as possible.
2. Draw as general figures as possible.
Model proofs
Work through Ex. 19 and 20 (analytic) and 21 and 22 (synthetic) on pages 65–66. Your student draws each figure and answers every "(Why ?)" aloud.
Ex. 21. The middle point of the hypotenuse of a right triangle is equidistant from the three vertices.
The meeting points of a triangle
Exercises 24–27 prove that the angle bisectors, the perpendicular bisectors, the altitudes and the medians of a triangle each meet in a point. Draw a large triangle for each and construct the lines carefully: they really do meet.
Ex. 27. The medians of a triangle meet in a point which is two thirds of the distance from each vertex to the middle of the opposite side.
Wrap up
Your student explains, in a sentence each, the synthetic, analytic and indirect methods.
Afterwards
Spread Exercises 19–27 over two sessions if needed. They are the best preparation for the original exercises that follow.
Answers to the book's problems
These are worked in the book; the "(Why ?)" answers are references to earlier sections (for Ex. 21, the line DE is parallel to AC by § 104 and bisects CB by § 188; D is then on the perpendicular bisector, § 160).
Worked out for this site from the scan. If a number in your copy differs, trust the book.