Wentworth's Plane and Solid Geometry · Book I, Props. XXIII–XXV · Book 2: Book I: Rectilinear Figures
12. Isosceles triangles, three sides, and right triangles
Goal: Your student can prove the converse of the isosceles theorem, the side-side-side case, and the hypotenuse-leg case for right triangles.
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 XXIII–XXV (§§ 147–151) with corollaries, and Exercise 8.
The lesson, step by step
Warm up
Ask: State Props. XX, XXI and XXII. Which one did we use to prove XXII?
The new idea
If two angles of a triangle are equal, the sides opposite the equal angles are equal, and the triangle is isosceles.
Two triangles are equal if the three sides of the one are equal, respectively, to the three sides of the other.
Two right triangles are equal if a leg and the hypotenuse of the one are equal, respectively, to a leg and the hypotenuse of the other.
Draw the figure
For Prop. XXIV: draw triangle ABC, then a copy A'B'C' turned over and placed against it so that the longest sides coincide, with B and B' on opposite sides. Join BB'.
Walk through the proof
XXIII: draw AD perpendicular to BC. The two right triangles share AD and have equal acute angles, so they are equal (§ 142), and AB = AC.
XXIV: in the turned-over position, AB = AB' and CB = CB', so each little triangle with base BB' is isosceles; the base angles are equal (Prop. XXII). Add them: angle ABC = angle AB'C. Now Prop. XXI applies.
XXV: put the triangles together along the equal legs so the right angles form a straight line. The two hypotenuses make an isosceles triangle, so the angles opposite are equal, and § 141 finishes it.
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. 8. How many degrees are there in each of the acute angles of an isosceles right triangle ?
Wrap up
Your student lists, from memory, all the ways so far of proving two triangles equal (§§ 139–143, 150, 151).
Answers to the book's problems
Ex. 8: 45° each.
Worked out for this site from the scan. If a number in your copy differs, trust the book.