Wentworth's Plane and Solid Geometry · Book I, Props. XIX–XXII · Book 2: Book I: Rectilinear Figures
11. Sides of a triangle, and two ways triangles are equal
Goal: Your student can prove that two sides of a triangle exceed the third, the angle-side-angle and side-angle-side cases of equal triangles, and that base angles of an isosceles triangle are equal.
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)
- Scissors and scrap paper
What's in the book
Propositions XIX–XXII (§§ 138–146) with corollaries, the note on the Principle of Duality, and Exercise 7.
The lesson, step by step
Warm up
Ask: What is the sum of the angles of a triangle? What is an exterior angle equal to?
The new idea
The sum of two sides of a triangle is greater than the third side, and their difference is less than the third side.
Two triangles are equal if two angles and the included side of the one are equal, respectively, to two angles and the included side of the other.
Two triangles are equal if two sides and the included angle of the one are equal, respectively, to two sides and the included angle of the other.
In an isosceles triangle the angles opposite the equal sides are equal.
Draw the figure
For Props. XX and XXI: cut out two equal paper triangles, ABC and DEF, with the matching parts marked.
For Prop. XXII: draw an isosceles triangle ABC with AB = AC (two equal radii from A with the compasses), and draw AD bisecting the angle at A.
Walk through the proof
XIX: AB + BC > AC, because a straight line is the shortest path; take BC from both sides for the difference.
XX: place AB on DE. The equal angles make AC fall along DF and BC along EF, so C lands on F (two lines meet in only one point).
XXI: place AB on DE. The equal angle makes AC fall along DF, and since AC = DF, C lands on F.
XXII: the bisector AD makes two triangles that are equal by Prop. XXI (two sides and the included angle), so angle B = angle C.
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.
Use the paper triangles to act out the "placing" in XX and XXI.
Exercises
Ex. 7. If the equal sides of an isosceles triangle are produced, the angles on the other side of the base are equal.
Hint: these angles are supplements of the equal base angles.
Wrap up
Read the note on the Principle of Duality (end of § 144). Ask your student to swap "sides" and "angles" in Prop. XX and see that it becomes Prop. XXI.
Tip
Paper cut-outs make the method of superposition concrete. Use them freely in this part of Book I.
Answers to the book's problems
Ex. 7: each angle beyond the base is the supplement of a base angle (§ 86); supplements of equal angles are equal (§ 85).
Worked out for this site from the scan. If a number in your copy differs, trust the book.