Are there closed plane figures that are not triangles that have an angle-sum of 180°?

[bphil]

Are there closed plane figures that are not triangles that have an angle-sum of 180°?

Hi guys,

I have the following question: are there closed plane figures that are not (rectilinear) triangles that have an interior angle-sum of 180°?

Or, the other way round: if I know that X has an interior angle sum of 180° am I entitled to think that X is a (rectilinear) triangle?

Restriction: I'm only interested in objects of Euclidean geometry.
Possible answer I have in my mind but don't know how to further investigate: there are objects with an internal angle sum of 180° that are not (rectilinear) triangles. These objects are constructed with (at least one) non-straight line.

Can you give me any pointers? Thanks.

 

[Deleted member 4993]

Hi guys,

I have the following question: are there closed plane figures that are not (rectilinear) triangles that have an interior angle-sum of 180°?

Or, the other way round: if I know that X has an interior angle sum of 180° am I entitled to think that X is a (rectilinear) triangle?

Restriction: I'm only interested in objects of Euclidean geometry.
Possible answer I have in my mind but don't know how to further investigate: there are objects with an internal angle sum of 180° that are not (rectilinear) triangles. These objects are constructed with (at least one) non-straight line.

Can you give me any pointers? Thanks.

If you are not confined to rectilinear sides - 3 arcs of circles can form a closed figure without having the sum to be 180°. Once we get into arcs - the possibilities are limitless. It can be part of an ellipse, parabola, logarithmic spiral. I guess, we need to topology to find some rules for that world.

 

[bphil]

If you are not confined to rectilinear sides - 3 arcs of circles can form a closed figure without having the sum to be 180°. Once we get into arcs - the possibilities are limitless. It can be part of an ellipse, parabola, logarithmic spiral. I guess, we need to topology to find some rules for that world.

Thanks for your answer! Just to clarify, since I asked about the existence of plane figures with an internal angle-sum of 180° and you wrote that "3 arcs of circles can form a closed figure without having the sum to be 180°" and "Once we get into arcs - the possibilities are limitless": so it is true that there are figures (made up of arcs) that have an internal angle-sum of 180°? And those figures might very possibly consist of more than three intersecting lines?

 

[JeffM]

If you confine your attention to closed figures made from straight lines, then the triangle is the only one with the sum of its angles equaling 180 degrees.

 

[bphil]

If you confine your attention to closed figures made from straight lines, then the triangle is the only one with the sum of its angles equaling 180 degrees.

You guys have a funny way of answering Usually I'm fine with working out implications of statements, but since I really don't know enough about mathematics, I don't want to take chances. So as before: thanks for your reply, JeffM. Can I take from your response that if I do not restrict myself to straight lines, there are closed plane figures constructed from three (or more) lines that do have exactly 180° interior angles?

And, I would be grateful if someone could confirm this separately: are there closed plane figures constructed from more than three lines that have an interior angle-sum of 180°?

Sorry for harassing you guys for a yes or no-answer although from your perspective probably everything has been said, but again, I need to make sure. It would also be great if you someone could point me to some literature concerned with my question. Any ideas?

Thanks,
bphil

 

[JeffM]

You guys have a funny way of answering Usually I'm fine with working out implications of statements, but since I really don't know enough about mathematics, I don't want to take chances. So as before: thanks for your reply, JeffM. Can I take from your response that if I do not restrict myself to straight lines, there are closed plane figures constructed from three (or more) lines that do have exactly 180° interior angles?

And, I would be grateful if someone could confirm this separately: are there closed plane figures constructed from more than three lines that have an interior angle-sum of 180°?

Sorry for harassing you guys for a yes or no-answer although from your perspective probably everything has been said, but again, I need to make sure. It would also be great if you someone could point me to some literature concerned with my question. Any ideas?

Thanks,
bphil

I thought I had answered your second question. In the Euclidean plane, there is only one closed rectilinear figure where the sum of the interior angles is 180 degrees, namely the triangle. In fact, that is one way to define the Euclidean plane.

Once you escape the confines of Euclidean geometry. a "triangle" does not have an angle sum of 180 degrees. Example. Draw a perpendicular north from the equator along the spherical surface. Go one quarter of the way around the surface of the globe along the equator and draw a second perpendicular north along the surface. Those two perpendiculars will meet at the North Pole and form an angle of 90 degrees. There: you just drew a "triangle" with an angle sum of 270 degrees.

Aha you say, I was not asking about anything but plane figures. Fair enough. You plainly said "plane." But you did not tell us what you mean by "the" angle formed by the intersection of curvilinear lines and what restrictions (if any) you place on such lines. If we assume that the curves are smooth enough to have tangents and define the angle between them as the angle formed by their tangents at the point of intersection, a three sided closed figure may have an angle sum less than 180 degrees, equal to 180 degrees, or more than 180 degrees. In the absence of any further constraints, a six-sided closed figure, for example, may have an angle sum less than 180 degrees, equal to 180 degrees, or more than 180 degrees.

Edit: I think I just repeated what Subhotosh Khan said before.

 

[HallsofIvy]

An obvious example of a "non-triangle" having interior angles with angle sum 180 degrees is a semicircle sitting on a line segment. It has two angles, each 90 degrees.