HW 8
a) Give a list of steps with a carefully labeled diagram on how to use a compass and straightedge to find the bisector of a line segment on the plane. Your list of steps should include a proof that the construction process does in fact bisect the line segment.
b) Does your method work equally well on the sphere? Explain why or why not.
a) Give a list of steps with a carefully labeled diagram on how to use a compass and straightedge to find the bisector of any planar angle. Your list of steps should include a proof that the construction process does in fact bisect the angle.
b) Does your method work equally well on the sphere? Explain why or why not.
Given a small triangle and its three interior angles, derive the formula for the area of a small triangle. Include in your derivation the following:
A brief explanation for how you thought about adding up the various areas to get the initial formula
A brief explanation for how to conceptualize the derivation of the formula for the area of a lune
Appropriate illustrations to help clarify your explanations and derivations
Use your formula for the area of a small triangle to argue that the sum of the interior angles for a triangle on the sphere must be greater than 180 degrees and cannot equal 180.
Your derivation in number 3 was for a small triangle. Does this formula also work for the “shark fin” triangle? How about the “anti-triangle”? If yes, show how it applies. If no, explain why not.
HW 9
Derive a formula for the holonomy of a small triangle on the sphere. Include sketches and explanations with your derivation.
On a sphere, consider the parallel transports r and r’ along a great circle segment L. If r and r’ are extended (as great circles) in both directions then you will get a lune. Prove that L goes through the center of the lune. Be sure to give a definition for the center of the lune
Another way to state 2 is as follows: If a great circle transversal cuts a lune at congruent angles, then the transversal goes through the center of the lune.
Prove the converse of this statement. That is, prove that if a great circle transversal goes through the center of the lune then the transversal cuts the lune at congruent angles. State and use the same definition of center of the lune.
HW 10
Problems 1-3 are on the plane
Given straight line segments r and r’ are parallel transports along a straight line L. Prove that if you extend segments r and r’ as straight lines then they will never intersect. We will refer to this as the parallel transports are parallel theorem.
Prove this theorem using proof by contraction but do not use the fact that the sum of interior angles of a triangle equals 180 degree.
Prove this theorem, again by contradiction, but now use the fact that the sum of the interior angles of a triangle equals 180 degree.
2. The side-angle-angle (SAA) theorem states that given two triangles with side, angle, angle (in that order) congruent then the two triangles are congruent.
Prove SAA but do not use the fact that the sum of interior angles of a triangle equals 180 degree.
Prove SAA, but now use the fact that the sum of interior angles of a triangle equals 180 degree.
3. Prove that if two straight line segments r and r’ are parallel transports along another straight line L, then they are also parallel transports along any other straight line passing through the midpoint of the segment of L between r and r’.
Problems 4 and 5 are on the sphere
4. Is SAA true for all triangles on the sphere? If yes, prove it. If not, give a counterexample and determine a class of triangle (that is, a subset of all triangles on the sphere) for which SAA is true and explain why.
5. Prove that AAA is true for all triangles on the sphere
ANSWER FOR Q 5 HW 10
Using proof of contradiction: Assume that AAA congruence are not true for all triangles on the sphere.
Case 1. Given two spherical triangles with AAA congruence (diagram 2) but with different side lengths:
Line X’Y’ ≠ Line XY
Line X’Z’ ≠ Line XZ
Line Y’Z’ ≠ Line YZ
Figure 1. Two Triangle with AAA congruence
Diagram 1
Y’ and Y can be extended in one direction, and same as with Z’ and Z in another direction. This is true with parallel transport. This results to a formation of lune as seen in diagram 1. This entire lune has center at one direction (point P). But there exist two other centers within the lune due to the parallel transport: point a and a’. This then go against the logic that there should only be one center for lune. Therefore, this counterexample is of no use. By this contradiction, AAA congruence is true for all spherical triangles.
Case 2. With the given diagram below, there are two spherical triangles with AAA congruence with different side lengths:
Line X’Y’ ≠ Line XY
Line X’Z’ ≠ Line XZ
Diagram 2
Due to AAA congruence:
Angle XYZ = Angle X’Y’Z’
Angle YXZ = Angle Y’X’Z’
Y’ and Y can be extended in one direction, and same as with Z’ and Z in another direction. This is true with parallel transport. These lines will intersect at the point E labeled in the diagram shown which results to a formation of a lune with two centers (the point a and a’). Since a lune should only have one center, then this test is a contradiction.
One way to have only one center in the lune, angle Y’X’Z’ and angle YXZ should be more than 180 degrees. But of small triangle all interior angles must be more than 0 degrees but less than 180 degrees. Therefore, this contradiction is still invalid, and AAA congruence is still true for all triangles on a sphere.
In conclusion, AAA is TRUE for all spherical triangles because neither case 1 nor case 2 contradictions exist due to the theorems used in the proof.
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