Understanding Part (c) of the Protractor Postulate: Explaining r=30

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Discussion Overview

The discussion revolves around understanding part (c) of the Protractor Postulate in the context of a geometric model where points are defined as rational points in the Euclidean plane. Participants explore the implications of this model for the existence of points on a ray forming a specific angle, particularly focusing on the angle of 30 degrees.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant expresses confusion about the problem and seeks clarification on why the model does not satisfy part (c) of the Protractor Postulate.
  • Another participant interprets the model as involving points with rational coordinates and questions the existence of such points on the ray forming a 30-degree angle with the x-axis.
  • There is a repeated assertion that there are no rational points on the ray at 30 degrees, with a focus on the requirement that at least one point E must exist on the ray.
  • A participant emphasizes the need to prove the absence of rational points on the ray, indicating a deeper exploration of the implications of the postulate.
  • One participant expresses understanding after the discussion, suggesting that the conversation has clarified their confusion.

Areas of Agreement / Disagreement

Participants generally agree on the interpretation of the Protractor Postulate and the implications of the model involving rational points. However, there is an ongoing exploration regarding the proof of the absence of rational points on the ray, indicating that the discussion remains somewhat unresolved.

Contextual Notes

The discussion highlights the limitations of the model in satisfying the Protractor Postulate due to the specific nature of rational points and their geometric implications. The need for a proof regarding the absence of rational points on the ray is noted but not resolved within the thread.

pholee95
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I'm stuck on this problem. Can anyone please help me understand?

Consider the model of geometry where point means rational point in the Euclidean plane and all of our other terms have their normal interpretation. This model doesn't satisfy the Ruler Postulate because there isn't a one-to-one correspondence with R. It also doesn't satisfy part (c) of the Protractor Postulate. Explain why it doesn't satisfy this part of the postulate by considering the line through (0,0) and (1,0), the upper half-plane, and the number r = 30. (Hint: Use a little piece of trig and think about the point E in this case.)

*I know that part (c) of the protractor postulate states this: For each real number r, 0 < r < 180, and for each half-plane H bounded by AB there exists a unique ray AE such that E is in H and μ(angleBAE) = r◦.
 
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pholee95 said:
Consider the model of geometry where point means rational point in the Euclidean plane
I assume this means point with rational coordinates.

pholee95 said:
part (c) of the protractor postulate states this: For each real number r, 0 < r < 180, and for each half-plane H bounded by AB there exists a unique ray AE such that E is in H and μ(angleBAE) = r◦.
A ray is, first of all, a set of points. Are there any points with rational coordinates on the ray that forms $30^\circ$ with the $Ox$ axis?
 
Evgeny.Makarov said:
I assume this means point with rational coordinates.

A ray is, first of all, a set of points. Are there any points with rational coordinates on the ray that forms $30^\circ$ with the $Ox$ axis?

There are none right?
 
pholee95 said:
There are none right?
Right, except $(0,0)$. The axiom requires that at least one point $E$ lies on the ray, which is not the case here. But you have to prove that there are no rational points on the ray.
 
Evgeny.Makarov said:
Right, except $(0,0)$. The axiom requires that at least one point $E$ lies on the ray, which is not the case here. But you have to prove that there are no rational points on the ray.

Ah. I understand it now. Thank you so much for your help!
 

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