Wentworth's Plane and Solid Geometry · Book I, Props. XXVI–XXIX · Book 2: Book I: Rectilinear Figures
13. Unequal sides and unequal angles
Goal: Your student can prove that the greater side lies opposite the greater angle (and the reverse), and the two theorems on triangles with two sides equal but the third unequal.
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 XXVI–XXIX (§§ 152–155), the first proofs by the indirect method, and Exercises 9–12.
The lesson, step by step
Warm up
Ask: Name the cases in which two triangles are equal.
The new idea
If two sides of a triangle are unequal, the angles opposite are unequal, and the greater angle is opposite the greater side.
Reciprocally: If two angles of a triangle are unequal, the sides opposite are unequal, and the greater side is opposite the greater angle.
Prop. XXVIII (§ 154): with two sides equal, the triangle with the greater included angle has the greater third side. Prop. XXIX (§ 155) is its reciprocal.
Draw the figure
Prop. XXVI: draw triangle ACB with AB clearly longer than AC. On AB mark AE = AC and join EC.
Prop. XXVIII: draw two triangles sharing side AB, with BC = BE but angle ABC bigger; draw BF bisecting angle EBC and join EF.
Walk through the proof
XXVI: triangle AEC is isosceles, so angle AEC = angle ACE. Angle AEC is an exterior angle of triangle EBC, so it is greater than B. And angle ACB is greater than its part ACE. So ACB > B.
XXVII (indirect): AB is equal to AC, less than AC, or greater. Equal would make the angles equal; less would make C less than B. Both contradict the hypothesis, so AB > AC.
XXVIII: the bisector makes EF = FC. Then AF + FC = AF + FE > AE.
XXIX: indirect again, using XXI and XXVIII.
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.
Make sure your student can explain the indirect method: list every possibility and rule out all but one.
Exercises
Try Exercises 9–12 (pages 40–41). For example:
Ex. 11. If the vertical angle of an isosceles triangle is equal to 30°, find the exterior angle included by a side and the base produced.
Ex. 12. If the vertical angle of an isosceles triangle is equal to 36°, find the angle included by the bisectors of the base angles.
Wrap up
Ask: In a triangle with sides 5, 7 and 9, which angle is largest? (The one opposite 9.)
Answers to the book's problems
Ex. 9: 60° and 60° (whichever angle is 60°). Ex. 10: 48° 40' and 97° 20'. Ex. 11: 105°. Ex. 12: 108° (the angle facing the base; its supplement is 72°).
Worked out for this site from the scan. If a number in your copy differs, trust the book.