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Write-up for class on Tuesday, 9/22, by Jeff King
I. Various definitions of “small” triangle:
1. The triangle is contained in a hemisphere
2. The lengths of all sides are less than half the circumference of a geodesic.
3. The angles are less than a half turn
4. The sum of any two sides is less than the third side
5. The area of the triangle is less than half the surface area of the sphere
6. The triangle is contained in a quarter sphere
7. The sum of the interior angles is 180°
8. No single angle is greater than 180°
9. The triangle is contained in one-eighth of the sphere
10. The triangle is contained in an open hemisphere (one that does not contain the boundary)
11. Two sides do not contain antipodal points and the third side must be the shortest geodesic.
Each of these (after perhaps some tweeking) can be made to work.
We want to choose the least restrictive definitions of “small” triangle such that…
II. Thm: SAS is true on the sphere for all “small” triangles
…is true.
III. Thm: Angle-Side-Angle – If 2 angles and the side between them of two triangles are congruent,
then the triangles are congruent.
To show this, use isometries (translation, reflection, rotation) to lie on top of one another (see Figure
6.11, page 86 of textbook).
We also want to consider what happens on the sphere, and if ASA is true for all triangles or just for
“small” ones, as with SAS. To see this, draw a triangle with angle-side-angle as in Figure 6.12 in the text.
If you continue both of the outer geodesics all the way around the sphere, they will intersect twice,
forming two distinct triangles, rendering ASA false in general. Which restriction of “small” triangle
works for ASA? Is it the same as for SAS?
IV. The Triangle Inequality
It is fairly obvious that the triangle inequality, |a|+|b|>|c|, for a triangle with sides a,b,c, is true on the
plane (for proof, draw three segments, and draw circles about all three points with sides a, b, and c as
radii). But is it true on the sphere? Think of possible counterexamples (along the same lines of SAS and
ASA, “big” usually leads to bad). Is it true when restricting to “small” triangles?
Now think of another definition of the “triangle inequality” on the sphere, where we have
|a|S + |b|S > |c|S , where |a| S is defined as the shortest geodesic between the two endpoints of a.
Thus, |a| S is less than or equal half a geodesic circumference.
To show this is true, you only need convince yourself that any three points on a sphere lie in a
hemisphere. From this, and the restricted definition of triangle inequality, you should be able to
convince yourself that the modified triangle inequality is true. We will look at this more on Thursday.