Angle between 2 straight

RONY1 said:
Is there a link to a digital book in space geometry (neither trigonometry nor vectors) that includes claims / theorems and proofs?
In my language, there is no such book, nor paper / plain and digital ...
The above space geometry is neglected in my country's methodology
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I think solid geometry is somewhat neglected in my country as well! I have seen old high school books on solid geometry in used book sales, but the standard curriculum doesn't tend to include it, beyond measurement of solid figures. Part of what I was trying to ask you was, what have you been taught about solid geometry on which to base a proof? Evidently your point has been that you don't have that foundation.

If you want to study it, you might start by going through the part of Euclid's Elements that deal with the subject, namely Book XI.

Looking through that, I notice that he gives the definition of the angle between two planes that you are using (though you initially didn't state that the angles you use are perpendicular to the intersection line, which was what I was trying to draw out of you):

The inclination of a plane to a plane is the acute angle contained by the straight lines drawn at right angles to the intersection at the same point, one in each of the planes.​

As the source I'm linking to says, though,

As the previous definition requires certain assumptions, so does definition 6. It assumes that any two such acute angles are equal, something Euclid does not prove but could have in the course of Book XI.​

This is too common in Euclid! But see if you can use the propositions in this book to make the proof you want.

Looking for a more recent book, by searching for "solid geometry textbook", I find this 1919 book, for one:

www.gutenberg.org

Solid Geometry with Problems and Applications (Revised edition) by H. E. Slaught et al.

Free kindle book and epub digitized and proofread by volunteers.
www.gutenberg.org www.gutenberg.org

Theorem XVI (section 111, p. 30) and its corollaries amounts to what you are asking about; you may or may not be satisfied with its proof, depending on what axioms you accept.

Here is a 1912 book with similar content:


Theorem XX (section 673, p. 324) is your theorem, stated a little differently.

I hope this helps a little. I, too, wish there were more available about synthetic (axiomatic) geometry of solids; other approaches, such as vectors and analytic geometry, are certainly more popular today (though probably for good reason).
 
I uploaded the painting before I saw a last reference added..I will read it now
 
One of your perpendicular marks is in the wrong place, but, yes, that's what I said was obvious. More steps (or at least a statement as to the justification of the conclusion) would be needed to make a complete proof, depending on what theorems you have available.
 
JeffM said:
But two lines that share a point, define a plane. Just work in that plane. No need to bother with space or any other plane.
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The challenge is to prove that 2 angles are equal with each other on a different plane. Incidentally, in the simple plane geometry, angles between parallel lines - no mention of 90 degrees in the space.
In the plane geometry - everything is also based on one axiom
 
RONY1 said:
The challenge is to prove that 2 angles are equal with each other on a different plane. Incidentally, in the simple plane geometry, angles between parallel lines - no mention of 90 degrees in the space.
In the plane geometry - everything is also based on one axiom
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I don't know exactly what you meant by that. Plane geometry (Euclidean) is based on five axioms (Postulates):

1. A straight line may be drawn between any two points.
2. Any terminated straight line may be extended indefinitely.
3. A circle may be drawn with any given point as center and any given radius.
4. All right angles are equal.​
5. If two straight lines in a plane are met by another line, and if the sum of the internal angles on one side is less than two right angles, then the straight lines will meet if extended sufficiently on the side on which the sum of the angles is less than two right angles.​
 
I meant 5 ... but I just felt that it was involved. Anyway in space geometry there is a basic theorem # 1, which we are actually trying to prove: "2 angles in space whose shells are parallel to each other -equal Or that amount is 180 degrees
 
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I haven't yet had a look at the books in digital that have been offered to me here ... whatever they are in my language and i will use translation
 
Dr.Peterson said:
I think solid geometry is somewhat neglected in my country as well! I have seen old high school books on solid geometry in used book sales, but the standard curriculum doesn't tend to include it, beyond measurement of solid figures. Part of what I was trying to ask you was, what have you been taught about solid geometry on which to base a proof? Evidently your point has been that you don't have that foundation.

If you want to study it, you might start by going through the part of Euclid's Elements that deal with the subject, namely Book XI.

Looking through that, I notice that he gives the definition of the angle between two planes that you are using (though you initially didn't state that the angles you use are perpendicular to the intersection line, which was what I was trying to draw out of you):

The inclination of a plane to a plane is the acute angle contained by the straight lines drawn at right angles to the intersection at the same point, one in each of the planes.​

As the source I'm linking to says, though,

As the previous definition requires certain assumptions, so does definition 6. It assumes that any two such acute angles are equal, something Euclid does not prove but could have in the course of Book XI.​

This is too common in Euclid! But see if you can use the propositions in this book to make the proof you want.

Looking for a more recent book, by searching for "solid geometry textbook", I find this 1919 book, for one:

www.gutenberg.org

Solid Geometry with Problems and Applications (Revised edition) by H. E. Slaught et al.

Free kindle book and epub digitized and proofread by volunteers.
www.gutenberg.org www.gutenberg.org

Theorem XVI (section 111, p. 30) and its corollaries amounts to what you are asking about; you may or may not be satisfied with its proof, depending on what axioms you accept.

Here is a 1912 book with similar content:


Theorem XX (section 673, p. 324) is your theorem, stated a little differently.

I hope this helps a little. I, too, wish there were more available about synthetic (axiomatic) geometry of solids; other approaches, such as vectors and analytic geometry, are certainly more popular today (though probably for good reason).
Click to expand...
Accessing the book is great on you‏‏לכידה ד.JPGr part. I glanced at another page attached. I wonder what section there in the proof, probably based on the axiom.
 
That's about what I expected a proof to look like, written out in detail.

I'm not sure what you are asking about sections and axioms.
 
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