Math 4510 – Class Notes October 6, 2009
Anna Papush (with slight editing by DWH)
Problem 8.1 EEAT
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All angles < π/2 (true but too restrictive)
We want the theorem to work for other triangles as well
Consider the drawing of the counterexample in figure 8.3 on page 111.
Problem 7.3
b.) The sum of the angles of a planar triangle = π
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To prove this, we do not need to use properties of parallel lines, we can instead use the concept
of a sphere, or even a hyperbolic plane, of infinite radius
Hint: use 7.1
HS EAT (Exterior Angle Theorem):
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Not true on a sphere or on a hyperbolic plane
Follows easily from the sum of the angles = π
If α + β + γ| = π  then α + β = π – γ| = γ
Problem 6.4
SAS is true for a triangle on a sphere if:
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There are no antipotal points included in the triangle, not just restricted to only the vertices.
All sides of the triangle are each limited to less than the length of half a great circle (geodesic).
The triangle is contained within an open hemisphere.
All the angles in the triangle are less than a half-turn.
An important counterexample that tests the fourth proof would be a triangle with 2 antipotal vertices,
meaning there are an infinite number of possible geodesics between them. Then the third vertex of the
triangle lying on such a geodesic would have an angle of π.
In Thursday’s class, we are voting on 5 book chapters to work on after Chapter 8. Look over the rest
of the text and be ready to vote.
Chapter 8 – Parallel Transport
l1 is a parallel transport of l2 along l
Problem 8.2
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Think in terms of symmetries
Straight lines must intersect on sphere, but where?
On a plane, the lines will never intersect, but what about on a hyperbolic plane?
There is half-turn symmetry about central point P; the lines are equidistant and will never intersect (on a
plane). Adaptations of this:
Plane: if the lines intersected on one side, they would then diverge on the other side. Since there is halfturn symmetry about P, this must be impossible.
Hyperbolic Plane: lines can be asymptotic and never intersect, but not necessarily follow our notion of
parallel, depending on the definition. Despite the lines not necessarily being equidistant, they would still
be symmetric about a central point P -- ?but reflective symmetry as opposed to half-turn?
Sphere: there would still be half-turn symmetry about a central point, but not equidistant properties.