Hyperbolic triangles proof help?

In summary, in hyperbolic geometry, the accompanying figure shows M and N as the respective midpoints of AB and AC. The indicated angle measures are θ and ∅. Using AAA congruence for triangles and the fact that the internal angle sum of a triangle is less than 180 degrees, it can be proven that θ and ∅ are not equal to each other. The relationship between them can be determined with a proof, and it can be shown that either θ<∅ or θ>∅.
  • #1
bonfire09
249
0

Homework Statement



Prove in hyperbolic geometry: In the accompanying figure M and N are the respective (hyperbolic) midpoints of AB and AC and θ and ∅ are the indicated angle measures. Determine, with proof, which of the following is true:
(1): θ=∅ (2): θ<∅ (3): θ>∅ ( stands for phi)


Homework Equations


I have attached a picture of the figure too. Since I am in hyperbolic geometry I get to assume that the internal angle sum of a triangle <180 degrees and AAA congruence for triangles.


The Attempt at a Solution


I don't know where to get started. It looks simple but I am thinking that I need to get congruent triangles in there somehow but I don't know how to. If you can give me a hint that be great.
 

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  • #2
bonfire09 said:

Homework Statement



Prove in hyperbolic geometry: In the accompanying figure M and N are the respective (hyperbolic) midpoints of AB and AC and θ and ∅ are the indicated angle measures. Determine, with proof, which of the following is true:
(1): θ=∅ (2): θ<∅ (3): θ>∅


Homework Equations


I have attached a picture of the figure too. Since I am in hyperbolic geometry I get to assume that the internal angle sum of a triangle <180 degrees and AAA congruence for triangles.


The Attempt at a Solution


I don't know where to get started. It looks simple but I am thinking that I need to get congruent triangles in there somehow but I don't know how to. If you can give me a hint that be great.
Where's the figure?
 
  • #3
I've attached it.
 
  • #4
bonfire09 said:
I've attached it.
OK I see it now.

I guess that it can take a while to appear initially.
 
  • #5
The only things that I can really with is triangle defect and the the internal angle sum of triangles are less than 180 degrees. In case of quad's they are less that 360. But I am also given that M and N are midpoints and I know somehow they allow me to do something but not quite sure.
 

Related to Hyperbolic triangles proof help?

1. What is a hyperbolic triangle?

A hyperbolic triangle is a type of triangle that is formed on a hyperbolic plane, which is a non-Euclidean geometry. In a hyperbolic triangle, the angles and sides are not congruent to those in a Euclidean triangle.

2. How do I prove properties of hyperbolic triangles?

To prove properties of hyperbolic triangles, you can use the hyperbolic trigonometric functions, such as hyperbolic sine, cosine, and tangent. These functions are used to calculate the angles and sides of hyperbolic triangles.

3. What is the difference between a Euclidean triangle and a hyperbolic triangle?

The main difference between a Euclidean triangle and a hyperbolic triangle is the geometry in which they are formed. A Euclidean triangle is formed on a flat plane, while a hyperbolic triangle is formed on a curved plane. This results in different properties, such as the angles and sides, of the two types of triangles.

4. What is the significance of hyperbolic triangles in mathematics?

Hyperbolic triangles play an important role in non-Euclidean geometry, which is a branch of mathematics that studies geometries that do not follow the rules of Euclidean geometry. Hyperbolic triangles also have applications in fields such as physics and computer graphics.

5. How can I use hyperbolic triangles in my research or experiments?

If your research or experiments involve non-Euclidean geometry or curved spaces, then hyperbolic triangles may be useful in understanding and analyzing the data. Additionally, if your research involves fields such as physics or computer graphics, hyperbolic triangles may also be relevant.

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