Dilemma!

Geometry Level 3

Is the sum of angles in a triangle always 180 degrees?

Note: Consider all possible geometries.

No Yes

Ashish Nagpal by Ashish Nagpal , 24, India

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2 solutions

Daniel Liu
Jul 3, 2015

Hyperbolic Geometry has the angles of a triangle add up to less than 18 0 ∘ 180^{\circ} . Spherical geometry has the angles add up to more than 18 0 ∘ 180^{\circ} . So the answer is "No".

7 Helpful 0 Interesting 0 Brilliant 1 Confused

There can be geometries with no notions of angular measure, yet have coplanarity and parallelism.

See: http://www.maa.org/sites/default/files/pdf/pubs/SetsPlanetsAndComets.pdf

I could easily define a triangle here with the sum of all it's angles undefined

0 Helpful 0 Interesting 0 Brilliant 0 Confused

Moderator note:

Is there no way to define angles between lines on a projective plane? You might be able to get by with "angles between any two lines are a constant, no three lines lie in the same plane".

Note: It is true that there is no common notion of angular measure on the finite projective plane, because there are only finitely many points.

But I never mentioned snything about projective planes

Agnishom Chattopadhyay - 5 years, 11 months ago

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