# Talk:Angle

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## Classification, Identifying and Naming

I think there needs to be a clean up on classifying (obtuse, acute reflex) identifying and naming (∠ABC) categories. I will try to work on it when i find more time. —Preceding unsigned comment added by Vrkunkel (talkcontribs) 23:49, 20 April 2010 (UTC)

## Angle Conventions

Should there not be a section on angle conventions ie. that clockwise rotations are generally positive within the cartesian plane, and that angles are usually measured from the x-axis? —The preceding unsigned comment was added by 195.83.126.10 (talk) 09:32, August 20, 2007 (UTC)

Did you intend to write anti-clockwise (counter-clockwise in the US)? Dbfirs 09:06, 15 February 2009 (UTC)

## ERROR in the section Formal definition/using rotation

It is NOT true that there exists only one linear application sending vector u to vector v. I am not sure about the true statement needed here, something like "there exists only one positive isometry...". Can someone who knows better update that section? Thanks. Jeanot2432 18:36, 31 May 2006 (UTC)

## Alternate definition

How about defining angle as the ?linear? codependance/independance of two curves, 0 being identical and 1 being orthogonal? I think the whole difficulty is defining angle in such a way as to make complex angles a logical extension of real angles, while keeping the definition extremely simple and clear. Kevin 2003.03.14

## Angles in Complex Hilbert spaces

Here are some thoughts about angles in complex Hilbert spaces. I moved them from the main page because they don't qualify as encyclopedic knowledge. One could also use the absolute value of the dot product I suppose. --AxelBoldt

For complex Hilbert spaces, the formula (*) can be recycled to obtain a complex angle, but it is not entirely clear that this corresponds to a real-world notion of angle. An alternative is to use

(**) R(u·v)=cosθ ||u|| ||v||

where R denotes the real part. Definition (**) also special cases to (*) for real Hilbert spaces, so that may be a reasonable choice.

## Definition?

The definition of angle on the main page seems rather vague. Perhaps a better definition would be: the fraction of the arc of a circle with a center at the origin of the angle.

That way degrees can be clearly defined as:

(s/c)×360

and radians as:

(s/c)×2π = s/r

where s = arc length, c = circumphrence, and r = radius.

## Grad?

Scientific calculators often have a 3rd measure of angle for trig functions, abbreviated to "grad" , with 400 per circle -- what are they?

See gon. —Herbee 00:31, 2004 Mar 8 (UTC)
Often used in civil engineering in France, a gradian (or grad) is a unit of angle such that a right angle is 100 grad. While this may look smart, it stops people from dividing a right-angles into three equal parts... I think it was invented during the French revolution, when everybody thought it might be a good idead to count in tens rather than in twelves... A bit silly really, as 12 is dividible by 2,3,4 and 6 whereas 10 is only dividible by 2 and 5... The only reason we count in tens is because we have ten fingers, which proves the Babylonians were much more advanced than the French of 1789... Regards, -- Deimos 28 20:31, 18 March 2006 (UTC)

## Dimensionless?

[quote] Note that angles are dimensionless, since they are defined as the ratio of lengths. [/quote] Let us not confuse too many things here with this statement. Pi is defined by the ratio of two lengths. Pi can then be used to define the dimension of direction on a unit circle. Certainly, units are associated with things that have dimension. Angles have units, and they require the correct unit for a specific equation (Perhaps it is helpful to think of Radians as dimension-less and all other forms of angular units as dimensions.). Also, angles are a dimension, in polar and spherical coordinates, they have the dimension of direction. Angles depict two different directions.

(William M. Connolley 07:39, 2 Sep 2004 (UTC)) Angles are dimensionless but they may have units (degrees). Use of an angular coordinate as a dimension in one application isn't the same question. Direction is not a dimension.
User: Nobody_EDN 2004.10.22

Then why do the first three dimensions, length, width, and breadth, have three different directions??? 'Different directions' is the basic property of dimension. ---- Please tell me what definition of dimension you are using where an angle is ruled out. (They have units and they can define a point in space. Just as width, and breadth can, using length as the third dimension for either.)

(William M. Connolley 20:09, 23 Oct 2004 (UTC)) You can sign your comments by using 4 tildes ~~~~ like that. Now, on: I'm using "dimension" is the sese of dimensional analysis. Since angles are length/length (or at least thats one good way to define them) they have dimensions of L/L = no dimensions.

Pi is a ratio of two lengths.

(William M. Connolley 20:09, 23 Oct 2004 (UTC)) Yes, exactly. Pi is a pure number, and an angle. Hence, angles are dimensionless. Another way of seeing it.

Angles are defined by the vertex of part of a disk, cut once or more through the center. Angles are measured in fractions of Pi, or fractions of the full circle. Angle measurements are chosen by taking a full circle and cutting it up in pieces by a set constant. That constant is 2Pi for radians, 360 for degrees, 400 for gradients (grads), 800 percent for percent grade (Such that 45 degrees is a 100% grade.), and 32 for compass points.

Perhaps defining Pi, Circle, Circumference and Radius first would help in defining the Angle. Use a demonstration of the wrapping function.

A Circle, drawn with Radius of one unit, on a number line, centered at minus one, has zero located on the outer rim of the Circle. If the number line is then wrapped around the circle, the numbers one, two, three, four, five, and six will all be wrapped around the edge before we get back to the beginning of the circle where number zero is.

The number Pi is the ratio of Circumference over the Diameter, Or C/D. The Circumference over the Radius C/R is 2Pi. So the wrapping function has at 180 degrees, or the opposite side of the circle from zero, the number Pi. 2Pi coincides back again with the number zero. (A picture, or several, at this point would help.)

An angle is the position on the wrapped number line in Radians. Radians are in fractions and multiples of Pi. Pi is C/D and therefore length over length and is though of as dimension-less. Although they still can contain a dimension of direction at times, most mathematical and engineering equations eliminate any dimension from calculations. 2Pi times Radial length equals length of circumference..... 2Pi times Radius time rpm become velocity.....

Degrees are a different way of unitizing Radians. Degrees break up a circle into 360 degrees. Therefore, 2pi = 360 degrees. Once converted to Degrees the angle then carries the dimension of Degrees and all mathematical equations they are used in must deal with that dimension/unit, when doing dimensional/unit checks.

Other ways are points, grads, and percent. They carry the same dimension/unit requirement with them.

Supplied for creative incentive to improve the definition and description of an angle.

User: Nobody_EDN 2004.09.01

Asserting an angle is dimensionless because it is the ratio of two lengths is quite invalid, it merely establishes that it does not have the dimension of length, a claim not generally made. An angle is a measure of rotation in the same way that length is a measure of displacement and as such is a dimension. Rotation is susceptible to the same differential and integral operations as displacement, it is not some mathematicl mystery hidden away with obscure dimensionless properties. The sooner this article recognises these facts the better.--Damorbel (talk) 07:17, 5 June 2008 (UTC)

I'm in agreement with that. The fundamental unit of angle is one complete turnaround, ie. a circle has an angle "in it" of one complete turnaround. We usually measure angle in radians or degrees. A degree is explicitly 1/360 of a turnaround. A radian is 1/2pi of a turnaround, but it's considered dimensionless. Do trig functions like sin() have arguments of dimensionless number, or of angle? I'm not sure anybody has really thought this out or decreed a convention.Friendly Person (talk) 16:23, 3 May 2010 (UTC)
Yes, all trig functions are functions of real numbers (or complex in some treatments), and the values are just numbers. Mathematicians really have thought about this, and have agreed that angle does not have "dimension" in the sense used in Dimensional analysis, nor in the everyday sense of 2-D & 3-D. Are you thinking of "units" of angle (e.g. degrees, radians, grads)? You might be interested to read our article on Dimensionless quantity which includes radian as a measure of angle. Dbfirs 16:49, 3 May 2010 (UTC)

## Link suggestions

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## Reflex angle

Now, I'm certainly no math whiz... But do these exist? I googled for a diagram of such, but there was literally 0 search results, and the number one result for text-based material was this article. --69.132.195.212 18:07, 14 July 2005 (UTC)

Ack, sorry. This is User:Thorns among our leaves... just not logged in. --69.132.195.212 18:08, 14 July 2005 (UTC)
Uhm… this is my first hit: http://mathworld.wolfram.com/ReflexAngle.htmlGustavb 20:36, 3 March 2006 (UTC)

## Supplementary angles

Quote:

   * The difference between an acute angle and a right angle is termed the complement of the angle
* The difference between an angle and two right angles is termed the supplement of the angle.


Okay, so it is defined that you can only use an acute angle to compute the complementary. Does this mean though that for supplementary angles, you can only use either an acute angle or an obtuse angle? (effectively an angle in the range 0..180)

Is it illegal to ask, for example, for the supplementary angle of -10? The same question but for 190? --195.85.158.73 16:03, 10 March 2006 (UTC)

One could define a convention, but it would be valid only for a particular situation, and would not be generally agreed, so it is safest to avoid complement and supplement outside the basic range. Dbfirs 09:10, 15 February 2009 (UTC)

## sin(theta) = sin(theta+pi)?

Under the section "A formal definition" under the subsection "Using trigonometric functions" there is a statement that sin(theta)= ... = y/x = -y/-x = sin(theta+pi). That is not true. sin(theta) does not equal sin(theta+pi). As an example consider when theta = pi/2. sin(theta) = 1, while sin(theta + pi) = -1.

No, the statement is that sin(θ)/cos(θ)=y/x=(−y)/(−x) = sin(θ+π)/cos(θ+π). This is true because sin(θ+π)=−sin(θ) and cos(θ+π)=−cos(θ). Bo Jacoby 14:17, 10 July 2006 (UTC)

## Explementary Angles

Angles that sum to a full circle are explementary. (This continues the concept that angles that sum to a quarter circle are complemetary, and those that sum to a semicircle are supplementary.)

## Units

What's the name for the "clock face" measure of angle (e.g. where 12 o'clock is forwards, 9 o'clock is left, etc.)? Ojw 22:54, 23 August 2006 (UTC)

## People misuse the word "equal" for angles

Saying that two angles of equal measure are equal is not quite as clear and accurate as saying that they are congruent. For example, if I construct an equilateral triangle ABC and then let D be a point between B and C, I cannot say that angles DBA and DCA are equal under Leibniz's law, because for the predicate of whether they contain point B, their truth values are different.

Angles of equal measure are instead referred to as congruent. This is sufficient to describe them, because two Euclidean angles are congruent if and only if they are of equal measure.

In a Euclidean system, congruence is fundamental; it's the counterpart of an equals sign in numerical analysis.

In this article I have corrected a few uses of "equal" for angles, and in other articles I plan to do the same.

Benzi 17:00, 26 November 2006 (UTC)

In old books angles can be refer to as equal but in new books they can not. Zginder 20:05, 1 May 2007 (UTC)

Actually, "equals" is not a misuse. Both have a 90 degree angle. DBA and DCA above are triangles, if they were not triangles, then neither could contain a point in the first place. Consider, each makes a 90 degree angle, and obviously 90+90=180, as should be. However, DBA+DCA is not a number. It is a number, however, if you mean to say that DBA is not a set in the plane, but the angle formed; in which case, B is not in it as it is not a set. To be short, you are equivocating.Phoenix1177 07:11, 2 November 2007 (UTC)

a more meaningful explanation for the reason why 'equals' is a misuse would be by definition: an angle is defined either as a set of points of a plane limited by two rays with common endpoint including them, or just the rays, so an angle is a set regardless of the definition you choose, so the relation of equality between angles is the relation of equality between sets, and so it mustn't be confused with cogruence, as shown by benzi's example.--Ghazer (talk) 12:48, 11 February 2008 (UTC)

This is not essentially different from the common abuse of language when saying "two equal line segments" instead of "two line segments of equal measure". While somewhat sloppy, everyone understands the intended meaning.  --Lambiam 21:19, 11 February 2008 (UTC)
nonetheless, its still a mistake. wikipedia's aim is to be encyclopedic and thus accurate, and so ambiguities and plain mistakes should be avoided as much as possible - you could just as well say there's nothing wrong with replacing each 'they're' with 'their' & vice versa - most people would understand it, but would it be correct? no. common abuse is still an abuse.--Ghazer (talk) 14:27, 13 February 2008 (UTC)
There exist different definitions of what an angle is. Mathworld defines an angle as an "amount of rotation".[1] PlanetMath uses a notion of "free" angle for one in which (for example) all right angles are equal.[2] Mathematicians have no qualms in employing "common abuse of language" when it simplifies the discourse, as when they say "the function f(x)", as long as they are understood. This specific abuse is also rampant on Wikipedia, as are many other common abuses. Good luck in rooting it out. Personally I think it is more fruitful to explain that when angles are said to be equal, this usually and normally means they are equal in size.  --Lambiam 10:36, 14 February 2008 (UTC)
The use of the word congruent for angles in Euclidean geometry is confusing because the angles as drawn (with differing length of ray) are often not congruent in the normal mathematical sense, even when they are equal in size in the sense (isos, equal) used by Euclid. For this reason, as well as keeping the language simple, many mathematicians prefer to avoid the word congruent for angles. Dbfirs 09:22, 15 February 2009 (UTC)
The use of congruent for angles refers to the measure of angle rotation between the initial side and the terminal side. One of the earliest proofs introduced to new geometry students is to show that two right angles are congruent. That is true regardless of the length of their sides. Keep in mind we are referring to the angle of rotation. The length of an angle's side is rarely, if ever, of interest. Just as an aside, all the geometry teachers I've met insist upon using the word congruent for angles. JackOL31 (talk) 03:17, 18 October 2009 (UTC)
Similarly, the use of "equal" for angles refers to the measure of angle rotation between the initial side and the terminal side, and has done so for more than two thousand years if you count Euclid's "isos". All the geometry teachers I've met insist upon using the word equal for angles and would object to the use of congruent for angles drawn with different lengths of side. As I said elsewhere, we seem to be "divided by a common language". Dbfirs 09:42, 18 October 2009 (UTC)
I have addressed this on the Triangle Talk page so I'll not be pursuing it here. JackOL31 (talk) 22:38, 18 October 2009 (UTC)

Starting fresh. There is already a fairly reasonable explanation for the alternative (not lesser) definition for congruent angles on this page. However, I would change it to this: Angles that have the same measure (i.e. the same magnitude) are sometimes said to be congruent. Under this definition of congruent angles, an angle is defined by its measure only and is not dependent on the lengths of the sides of the angle (e.g. all right angles are congruent).

In the original explanation, much of the remaining text was simply negative spin. This explanation should stick to the topic of the alternative definition and not "here is an alternative definition and here is my smackdown". Of course, a fleshed out alternative angle definition may be required, depending on how you interpret the last line of the first paragraph. (see thesaurus.maths.org - an angle is a measure of turn) JackOL31 (talk) 21:57, 3 November 2009 (UTC)

I will make the above change in a week or so providing an opportunity for feedback prior to the change. JackOL31 (talk) 00:58, 7 November 2009 (UTC)
Yes, I think that it is fair to remove any negative spin, since the disputed definition of "congruent" is taught in many schools in the USA. We do need to make clear, however, for those not taught in the current American education system, that this does not mean that the diagrams representing the angles are congruent. Dbfirs 18:19, 7 November 2009 (UTC)
In my proposed change, I did try to state that we are only referring to an angle as a measure and not the angular figure itself. I didn't go into more detail since this is an article on angles rather than congruency and the update is for a small snippet within the article. If you have specific suggestions, I'd be happy to discuss them and revise as necessary. I believe what you're suggesting is what I was planning for the Congruence_(geometry) article, but I'm open to revision. Also, I was also thinking the link should also be changed to Congruence_(geometry) instead of the disambiguation page.
The proposed revision for congruent angles is reasonable given one uses the definition of angle as angular magnitude. The Congruence article essentially deals with congruence of "figures". However, most of the time we do not think of the term "angle" as a figure. We even say things such as "two sides and the included angle". Where some default to angle_figure, others default to angle_measure. Angles such as angle of inclination, elevation, depression, incidence, etc are angles not based on any length of sides. Same with angle settings for a mitre saw or surveying equipment. This alternate view avoids the implication that a quadrilateral has 4 angles and 8 sides (versus 4 angles and 4 sides). Also, it avoids the disconnect for similar, but noncongruent polygons having noncongruent angles when they both have corresponding angles with the same "opening" (we already know the corresponding sides are in proportion, all that is left are the angles). Both views have their pros and cons, but I wouldn't say it is a disputed definition. It is a different but equal definition. JackOL31 (talk) 22:33, 8 November 2009 (UTC)
Yes, the link needs changing. I don't think anyone will argue with that, so I'll go ahead and make the change. I also agree that most people think of "angle" as meaning "angle (measure)", which is why, in the UK, "equal" is still the standard usage. "Congruent" makes sense if "angle" is defined as the set of points on two intersecting infinite rays, which, presumably, is what is taught in the USA? Dbfirs 08:29, 9 November 2009 (UTC)
Yes, an angle is made up of two rays with a common endpoint. Angle congruency doesn't need infinite rays. That would be true for angle (figure), not angle (measure). The length of the sides do not affect an angle's measure. When we are discussing the measure, all we need is the "opening" or "the inclination of one line with respect to the other" to be congruent. In the US we use the standard "equal" (numeric) and the standard "congruent" (coincide). JackOL31 (talk) 02:11, 13 November 2009 (UTC)
I would have thought that angle congruency makes sense only when the infinite rays are included. If angle (measure) is meant, then the simple word "equal" (as used for more than 2000 years) is adequate and clearer. I don't understand "congruent (coincide)". Did you mean "superpositionable" as in the standard definition of congruent? Dbfirs 12:14, 13 November 2009 (UTC)
I believe we are still having an issue regarding figures as opposed to generated angles. Replace "measure" in my previous discussion with "inclination". Regardless of the lengths of the sides, are the inclinations the same? If yes, then the angles are congruent. As we discussed earlier, after doing some research I maintain that Euclid's equal is today's congruent. I further maintain that the word "equal" has not been used in that manner for the past 2000 years, except when one is concerned with the historical Euclid's Elements usage and when one is not mixing the two "equals". I would agree to the use of the current equivalent, "congruent". Please note that I will consistently reject an argument based on the misapplication (as I and others see it) of Euclid's "equal" with today's "equal". Back to your original question, since we are not dealing with numerical quantities we do not want to use numerical concepts. Further illustrated by the numerous exercises (proofs) from geometry textbooks that only deal with congruence. Yes, I was using shorthand for one item placed (superpositioned) on another to check whether they coincide. If you understand (not necessarily agree with) the concept of congruent angles having superpositioned inclinations (or slants, etc), then we should be good, yes? JackOL31 (talk) 21:50, 14 November 2009 (UTC)
You seem to be viewing this from the very restricted point of view some of modern American educators. The word "congruent" is just not used for angles elsewhere, and there are serious problems with its interpretation. The word "equal" satisfies all of your arguments for "congruent" and is much more widely understood. "Congruent" is certainly not the modern equivalent of "equal" to most mathematicians! Dbfirs 00:39, 15 November 2009 (UTC)

I see you ended the description of your last post with "...American view?" Let us go over once more the words from GimmeDanger. On a fundamental level, we are not here at Wikipedia to decide what is true or not, but rather to report what others have said about things. "Verifiability, not truth" is the catchy phrase. If there are alternate definitions for [you name it] in circulation, then those definitions should all be present on the [you name it] page. If there are alternate definitions for [you name it], those definitions need to be present. Wikipedia itself can't really take a stand on anything. Does that make sense? How many websites would you like me to reference for the particular definition of interest for congruent angles? How many texts? You should not be dismissing other views, American or otherwise. You are in the wrong if you do so. I disagree with your understanding and use of "congruent" and "equal". You seem to have an aversion to "congruent" and an affinity for "equal". Fine and dandy, but your preferences should not be allowed to affect these articles to the extent that the term "congruent" is removed. JackOL31 (talk) 02:51, 19 November 2009 (UTC)

Yes, I fully accept that some American educators use the word "congruent" for angles, and I have not attempted to remove the verifiable statement that you added to record this fact. We disagree on the usage of the word, and we will just have to agree to disagree, because educators disagree. I have never seen the word "congruent" used with this special meaning in any British text, and I can provide many references to support the use of "equal". If I seem to be taking an extreme view, I apologise. It was just in response to what I see as an extreme opposing view from the American side.

I am interested in you discussion above because I remain unhappy [3] with the definition of an angle as a figure.--Damorbel (talk) 10:05, 19 November 2009 (UTC)

I would be happy to define angle as an amount of turn, but there are alternative definitions. Dbfirs 19:54, 19 November 2009 (UTC)
Let's just hope that no one particular viewpoint holds sway over these articles and eventually expunges differing definitions or views. I just seems to me that this may have already happened by various like-minded, but well-intentioned editors. On another note, I don't understand your position of an extreme view. What would have been a substantively different definition for "angle" that you would have not held as extreme? I've checked the online OED and found, "angle - the space (usually measured in degrees) between two intersecting lines or surfaces at or close to the point where they meet." I then checked the Cambridge University Press and found, "Angle is a measure of turn. Angles are usually measured in degrees." The previous definitions come from British controlled publications, I believe. Following these definitions, I just don't see the congruent angles counterpart as an extreme position. I also don't see this as an American position. I just see it as a different mathematical position. I firmly believe this is where you and I differ vastly. On the "equal" point, my main concern is that I do not believe you fully understand the problem that that particular usage causes. Others learn that "equal" applies to numeric concepts, while "congruent" applies to nonnumeric concepts. If one is going to use "equal" as Euclid did, then one must note the distinction for those readers taught differently (i.e. substitute "congruent"). They will see that usage as mathematically incorrect. JackOL31 (talk) 03:50, 20 November 2009 (UTC)
An example of an "angle(figure)" would be any triangle with one side removed. Do you have another name for that shape? JackOL31 (talk) 03:50, 20 November 2009 (UTC)
Yes, I'm happy with your angle (figure) as a triangle with one side removed, but using this definition precludes the use of "congruent" in the sense agreed by all mathematicians since 1578 (possibly). The traditional meaning of congruent makes sense for angles only when angle is defined as a set of points bordered by two infinite rays meeting at the vertex. Dbfirs 08:11, 20 November 2009 (UTC)

The point of defining an angle as an "amount of turn" is that it eliminates the dimension of length from the definition, e.g. with a rotating sphere you don't need to know its size. More usefully the spherical coordinate dimensional method [4](and cylindrical) makes more sense when the angle(s) are seen to be real dimensions, not some construction derived from the radius (length), this makes for much clearer mathematics.

By the way, I am all in favour of the article mentioning common usage.--Damorbel (talk) 10:06, 20 November 2009 (UTC)

I'm happy to define angle as "amount of turn", but I'm puzzled by your dimension. Polar, cylindrical and spherical co-ordinates inherit their dimension from the radius co-ordinate. One can rotate for ever, but if one remains at the origin then no dimension is involved. I suppose it depends what we mean by "dimension". Dbfirs 16:24, 23 November 2009 (UTC)

In the Cartesian coordinate system (x y z) it is easy to imagine a coordinate being zero, the result is two-dimensional. In the cylindrical coordinate system if the angle is fixed or zero is it not the same? If one of height or radius is zero I think a polar coordinate system remains. In the spherical coordinate system the radius can be zero in which case we have a point, exactly as if x y z were all zero in the Cartesian coordinate system. Each system has useful symmetries but each can define all points in three-dimensional space.--Damorbel (talk) 17:34, 23 November 2009 (UTC)

Each of the coordinate systems needs a dimension of length to escape from zero. In Polar and Spherical co-ords, for example, the angle co-ords can take any value, but if the radius remains at zero then only a point is defined. In this sense, the angle values are dimensionless, but I agree that they add an extra dimension once the radius is allowed to increase from zero. Dbfirs 22:12, 23 November 2009 (UTC)

"Each of the coordinate systems needs a dimension of length to escape from zero." An amount of turning is not changed by the length of a radius because the three coordinate systems I refer to define mutually independent dimensions, they are orthogonal meaning (crudely) changes in one dimension do not influence anything in the others. Turning (revolution, angle) does not need length to define it, otherwise you would have a two dimensional coordinate system.--Damorbel (talk) 09:26, 24 November 2009 (UTC)

Yes, but what orthonormal basis would you use to generate R3 using spherical co-ords? If the radius vector is zero, only a point can be generated. Dbfirs 15:22, 24 November 2009 (UTC)

A vector defined in spherical coords is not substantially different from one in cartesian or cylindical coords, see here List of canonical coordinate transformations, the only concern is the complexity of the relevant expressions, try and imagine how you would navigate your boat in cylindrical or cartesian coords. in spherical coords you (usually) treat the radius as a constant so that angles measured with a sextant and time determined with a chronometer can be converted directly into distance etc. etc. It really would be an unnecessary sweat to try to do this with x y z or r h θ but it would not change the validity of your calculations.--Damorbel (talk) 20:44, 24 November 2009 (UTC)

I hope I am still on a track that is helping. There area a vast number of coordinate systems used in situations where they have a simplifying effect, usually from some underlying symmetry. --Damorbel (talk) 21:06, 24 November 2009 (UTC)

Yes, I'm well aware of the conversion equations, but the point I was making is that in (for example) spherical co-ords, one cannot define an orthonormal basis for R3 without each vector of any such basis inheriting its dimension from the radius vector. I agree that in the restricted case of navigation on the surface of a sphere of fixed radius, one can treat each angle as an independent dimension. Dbfirs 08:07, 26 November 2009 (UTC)

"Each of the coordinate systems needs a dimension of length to escape from zero." Why does it have to escape from zero? The concept of zero is valid in x y z coordinates, the (none WIKI) point of view is different with r h θ and r θ φ but these systems are not otherwise different.--Damorbel (talk) 08:57, 30 November 2009 (UTC)

In Cartesian coordinates, each base vector carries its own dimension of length, and, for example, if one takes the space generated by all independent multiples of any two base vectors, one generates a 2-D space. In the case of spherical coordinates, the space generated by all independent multiples of the angles generates only a single point (the origin). One has to include a non-zero multiple of the radius coordinate to escape from the origin. Whilst it is easy to specify an orthonormal basis using spherical coordinates, each member of this basis must necessarily include a radial component, thereby inheriting its dimension from that of the radius vector. Dbfirs 22:31, 30 November 2009 (UTC)

What I am suggesting is an alternate usage definition that the young reader on up to high school student can understand (heck, throw in the common person on the street). A simple definition written in the same vein as the CUP definition: Angle is a measure of turn. A definition to explain its usage in pre-collegiate math/geometry classes where angle refers only to the angle measure and not to the angle sides. Also, to explain how the word is used in triangle congruency postulates and in CPCTC/CPCFC. Similarly, its usage in concepts such as angle of inclination, elevation, depression, incidence, etc. JackOL31 (talk) 20:50, 12 December 2009 (UTC)

Yes, I agree that this should come first, with the "metaphysical" discussion of the nature of angle coming later. Dbfirs 00:30, 13 December 2009 (UTC)

## Old-school definition?

Just wondered if anyone has though of adding the old-school definition in? When I was in school all those years ago, we were taught to never refer to angles as things meeting at a point, but "when a line rotates about one of it's extremities in one plane, from one position to another.", in the trigonometric sense anyhoo :-) 15:53, 28 May 2007 (UTC)

Thinking of old-school, I recall my headmaster objecting to any definition beginning "when"! An angle is much more general than a line rotating about it's extremity, but I agree that's a good way to introduce the concept. Dbfirs 07:02, 11 September 2011 (UTC)

## Unnecessarily complicated?

Does anyone else think that the introduction of this "k" throughout the "Units of measure for angles" section is unnecessarily complicated? In my view it's just likely to totally confuse people. Matt 20:23, 30 June 2007 (UTC).

Well, I have rejigged this section now. Matt 19:47, 18 July 2007 (UTC).

## Semi Angle

Any chance of adding Semi Angle (e.g. Half the angle of the tip of a cone)- the only place I could find it was in my University Journal collection- not all web users have access to journals... —Preceding unsigned comment added by 124.181.109.145 (talk) 03:51, 16 September 2007 (UTC)

Not sure if this is very common? Probably not worth adding unless it is in standard textbooks. -- Steelpillow 06:57, 16 September 2007 (UTC)
Much more common than oblique angle (see next) in the UK. Dbfirs 06:58, 18 July 2009 (UTC)
Oblique angles are introduced on page 11 in the Honors Geometry textbook for my school district. Semi Angle is not mentioned. JackOL31 (talk) 03:27, 18 October 2009 (UTC)
I can find you school textbooks where semi-angle is mentioned, but I would struggle to find one in the UK that mentioned obliqe angle. However, since the term is in common usage in the USA, I'm sure that British readers can cope with learning a new meaning for oblique. Also, since semi-angle is used mainly just for cones and isosceles triangles, I'm not sure how important it is to mention it here. Should I add a very brief note? Dbfirs 00:48, 13 December 2009 (UTC)

## OBLIQUE ANGLE

suggestion: ADD oblique angle definition to article.

I did not see a definition for "oblique angle" in the article. —Preceding unsigned comment added by 199.80.66.119 (talk) 19:13, 29 January 2008 (UTC)

Is this usage common in the USA? Is is a useful definition? I've never heard it used in the British educational system. Dbfirs 06:58, 18 July 2009 (UTC) Dbfirs 06:58, 18 July 2009 (UTC)

## Reflex Angles

When we say angle ABC, how do we know that it refers to the acute/obtuse angle between AB and BC? Surely, there is always a reflex angle there too? What if we wanted to refer to that instead? —Preceding unsigned comment added by 81.159.24.29 (talk) 23:18, 7 February 2008 (UTC)

We say "reflex angle ABC". Dbfirs 09:26, 15 February 2009 (UTC)

## Angle is not a figure

An angle is certainly not a figure! Using the term "ray" is associated with Hilbert's modification of Euclidian geometry, much can be found in this link http://www.libraryofmath.com/.

A useful definition of an angle for the non mathematitician is that an angle is "a measure of turning". Definitions based on set theory tend to have difficulty with infinities and zeros (empty sets) that tend to arise with day to day mathematics so I think it is advisable that set theory considerations should be dealt with by references for further study instead of in the article.

The statement "Angles are considered dimensionless" has no basis in any mathematics, a brief consideration of spherical coordinates http://en.wikipedia.org/wiki/Spherical_coordinates should dispel any doubts (if anyone is in doubt spherical coordinates define points in three dimensional space by two angles and a length). Justification by a reference to dimensional analysis http://en.wikipedia.org/wiki/Dimensional_analysis is completely irrelevant. Dimensional analysis is a sort of checksum for consistency in formulae so that one avoids defining length by seconds or amps or other blush causing idiocies. As yet I haven't analysed the article sufficiently well to find the consequences of this whopper, they may well be profound, if someone else sees what is wrong and want to revise I will be very pleased, but I have never rewritten a Wiki article before.--Damorbel (talk) 20:54, 30 May 2008 (UTC)

The source you mention (Library of Math) defines an angle as a subset of the Euclidean plane, namely as the union of two rays, themselves sets of points.[5]. It has this to say about its measure:
Each angle A B C is associated with a unique real number between 0 and 180, called its measure and denoted angle measure mA B C. No angle can have measure 0 nor 180.[6]
This contradicts everything you write. Angles are figures, and the measure is a real number, which is dimensionless. You'll have to come up with other sources if you want to add alternative definitions. The definitions given at LOM (which, by the way, does not fit our definition of "reliable source") are perfectly fine for Euclid-style geometry.  --Lambiam 16:20, 31 May 2008 (UTC)

You maintain that an angle is a figure and that a source is required for alternative definitions. Do you have a source for your "figure" definition? After all the word "figure" has a number of meanings.

Saying that "This contradicts everything you write." is rather general, paricularly when you cite "No angle can have measure 0 nor 180". My understanding is that this kind of statement has origins in set theory, set theory tries to constrain mathematics by logic, a constraint which is OK if you confine your definition of mathematics to what a digital computer does, unfortunately mathematics got there long before computers, the upstart should not be putting restrictions on the senior subject. --Damorbel (talk) 11:56, 5 June 2008 (UTC)

I'm not maintaining any particular notion of what an angle "is". I'm happy to maintain, though, that the certainty expressed in the emphatic statement "An angle is certainly not a figure!" is baseless. There are various ways of defining the notion of angle, which is reflected in the article.  --Lambiam 07:17, 6 June 2008 (UTC)

One moment you say "The definitions given at LOM (which, by the way, does not fit our definition of "reliable source"), the next you cite "If you want a source for the definition of angle as a figure, well, much can be found in this link: http://www.libraryofmath.com/." which is of course LOM. Is this a variable source or a source dependent on who cites it?

You claim an angle is a figure, then you maintain an angle is dimensionless. But a figure, of whatever sort, requires the dimension length for its very existence. While I agree that an angle does not have length as a dimension, it seems to me to be wildly inconsistent to define it with a representation wholely dependent on length.

The current article even says "the angle is the "amount of rotation" that separates the two rays". "Separates" is imprecise, it can mean displaced as well as inclined. And the use of "two rays" is inconsistent when used to define a dimension that has no length. Curves can be rotated through an angle, it's happening to solids all the time! But for rotation there is no requirement for any representation of length. Similarly definitions based on a "vertex" are equally inappropriate.

To quote "I'm not maintaining any particular notion of what an angle "is" ". I do think an encyclopedia should give the clearest notion (let us say idea) of the subject of the article's title. In the case of an angle it is important that it is free of all connections with length is very important, witness polar coordinates http://en.wikipedia.org/wiki/Polar_coordinate_system and spherical coordinates http://en.wikipedia.org/wiki/Spherical_coordinate_system--79.79.30.151 (talk) 20:43, 6 June 2008 (UTC)

1. I was responding to Damorbel – I don't know if you are the same editor – and I just gave the same source this editor mentioned in support of the statement that an angle is not a figure. For a reliable source, see the reference given in the article itself: Sidorov, L.A. (2001), "Angle", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4.
2. I didn't write that an angle is dimensionless, but that the measure of an angle is dimensionless. However, as the article explains, the term "angle" is used interchangeably for both the geometric configuration itself and for its angular magnitude (which is simply a numerical quantity).
3. If several non-equivalent definitions of some concept are current, then, in my opinion, we should present all, and not just the "clearest".
4. I don't understand why you think that a figure, of whatever sort, requires the dimension length for its very existence. A single point is also a (very simple) figure, as is a line. Neither has a length. I also don't understand why you ascribe a particular significance to the use of angles in spherical coordinate systems, and why that should be incompatible with the present article.
--Lambiam 02:48, 7 June 2008 (UTC)

In geometry and trigonometry, an angle (in full, plane angle) is the figure formed by two rays sharing a common endpoint, called the vertex of the angle (Sidorov 2001). The magnitude of the angle is the "amount of rotation" that separates the two rays, and can be measured by considering the length of circular arc swept out when one ray is rotated about the vertex to coincide with the other (see "Measuring angles", below). Where there is no possibility of confusion, the term "angle" is used interchangeably for both the geometric configuration itself and for its angular magnitude (which is simply a numerical quantity).

The word angle comes from the Latin word angulus, meaning "a corner". The word angulus is a diminutive, of which the primitive form, angus, does not occur in Latin. Cognate words are the Latin angere, meaning "to compress into a bend" or "to strangle", the Greek ἀγκύλος (ankylοs), meaning "crooked, curved," and the English word "ankle." All three are connected with the Proto-Indo-European root *ank-, meaning "to bend" or "bow" —Preceding unsigned comment added by Gon56 (talkcontribs) 14:11, 13 June 2008 (UTC)

I've restored these two paragraphs because they were removed without explantion by an anonymouys editor. I don't think they are particularly good as an introduction, but are they false? Dbfirs 12:09, 13 June 2009 (UTC)

## Dimensionless-ness revisited

First, let me express relief that this article counters the often-seen claim that angles (or radians) are "unitless". Now, as to dimensionlessness:

Earlier in this discussion page, William M Connolley asserts the commonly seen position:

"Angles are dimensionless but they may have units (degrees). Use of an angular coordinate as a dimension in one application isn't the same question. Direction is not a dimension."

Damorbel attempts to rebut, later including the example of polar and spherical coordinates to show that angle can play the role of dimension.

The issue here appears to revolve around the meaning of "dimension", and WMC attempted to clarify his definition by saying:

"I'm using "dimension" in the sense of Dimensional analysis. Since angles are length/length (or at least thats one good way to define them) they have dimensions of L/L = no dimensions."

Yet on the dimensional analysis page there is recognition that you can account for dimensions in a coarser- or finer-grained manner, leading to "Extensions" and "directed dimensions". This seems to address the bone of contention here. (WMC says "Direction is not a dimension." -- right, but direction may be used to distinguish dimensions.)

In one scenario we may be happy to take the coarse view that angle is derived from two lengths which we don't care to distinguish, then OK, angle comes out dimensionless.

In a different scenario we might want to analyze dimensions more finely, and make a distinction between displacements in two orthogonal directions. In that case angle is derived from two distinct dimensions which do not cancel out (though their magnitude reference quantity, meter, does indeed cancel).

So, I propose, it's not a matter of whether angle is or is not dimensional. It's a matter of whether for a particular purpose we are keeping track of angle as a separate dimension, just as we might choose to keep track of x, y and z dimensions separately. Gwideman (talk) 03:04, 17 March 2010 (UTC)

When angle is used to keep track of dimensions, as in polar and spherical co-ordinates, it inherits its dimension from the essestial unit of distance in the co-ordinate system. Angle without a distance cannot fix positions in any space, so pure angle is dimensionless. Dbfirs 08:12, 17 March 2010 (UTC)
I think this conflates two or three different meanings of "dimension". As a challenge to the last sentence: How is this different than arguing that in 3-space, x and y alone can't fix a position, hence must not be dimensions? Gwideman (talk) 07:49, 21 March 2010 (UTC)
x and y alone fix position in 2-D space. Angle doesn't fix position in any space except (for example) the surface of a pre-defined sphere, where the dimension comes from the radius. Dbfirs 13:16, 21 March 2010 (UTC)
Radians are clearly a ratio, other measures are scaled ratios eg degrees, and have no corresponding physical dimension (eg mass, length, time, etc) - if some clarification is needed on what is meant by "dimensionless" that probably needs to be done on a more general article. In general I would expect most people to be thinking of "physical dimensional analysis" rather than in terms of degrees of freedom when the term "dimensionless" is used.Shortfatlad (talk) 13:46, 17 March 2010 (UTC)
I believe this misses my point. Sure, angle can be conceived of as a ratio pf arc length to radius. The question is whether these two lengths are of the same physical quantity, or different, which depends on the application (I claim). SI takes the position that length in any direction is the same quantity, hence is partially blind to angle and rotation, and winds up with angle and solid angle having units reducible to the same unit ("1"), and related problems with all the per-unit-angle and rotation-related quantities. Anyhow, as I am learning, this topic has a long history of debate, and I suppose we should refrain from reproducing it all here :-). Gwideman (talk) 07:49, 21 March 2010 (UTC)

## Not Equivalence Relation

I've moved this paragraph here because of the recent comment [] about non-transitivity. How much should we "rescue" and put back in the article? Dbfirs 02:23, 14 September 2010 (UTC)

### Using rotations

Suppose we have two unit vectors $\vec{u}$ and $\vec{v}$ in the euclidean plane $\mathbb{R}^2$. Then there exists one positive isometry (a rotation), and one only, from $\mathbb{R}^2$ to $\mathbb{R}^2$ that maps $u$ onto $v$. Let r be such a rotation. [Paragraph reads OK up to this point, then needs to be re-thought. For example $\vec{a}\mathcal{R}\vec{b}$, as defined, is not a transitive relation, so not an equivalence relation.] Then the relation $\vec{a}\mathcal{R}\vec{b}$ defined by $\vec{b}=r(\vec{a})$ is an equivalence relation and we call angle of the rotation r the equivalence class $\mathbb{T}/\mathcal{R}$, where $\mathbb{T}$ denotes the unit circle of $\mathbb{R}^2$. The angle between two vectors will simply be the angle of the rotation that maps one onto the other. We have no numerical way of determining an angle yet. To do this, we choose the vector $(1,0)$, then for any point M on $\mathbb{T}$ at distance $\theta$ from $(1,0)$ (on the circle), let $\vec{u}=\overrightarrow{OM}$. If we call $r_\theta$ the rotation that transforms $(1,0)$ into $\vec{u}$, then $\left[r_\theta\right]\mapsto\theta$ is a bijection, which means we can identify any angle with a number between 0 and $2\pi\,$.

## arrangement of Units

I'm thinking of rearranging the units in order of size, biggest first:

Any objection? —Tamfang (talk) 20:28, 8 May 2011 (UTC)

Seeing none ... —Tamfang (talk) 20:05, 29 June 2011 (UTC)

## Angles greater than a full turn

Is there a special term for angles greater than a full turn (360 degrees?) Reflex means angles between 180 and 360, so what do we call angles over 360? Degenerate angles? Redundant angles? Surplus angles? 173.165.239.237 (talk) 15:40, 24 May 2011 (UTC)

See winding number for a common expression of multiple turns about a point.Rgdboer (talk) 19:57, 3 November 2011 (UTC)
An angle of 360° is a turn (geometry). Then an arbitrary positive angle amounts to so many turns and a residue angle that is less than a full turn.Rgdboer (talk) 00:43, 4 November 2011 (UTC)

## Angle function?

The angle notation used in electronics and physics suggests an angle function which is more useful than the tangent function for the determination of angles. Consider the point of a vector (x,y) in a plane and,

$r=\sqrt{x^2+y^2}$
$\sigma=\begin{cases}-1\quad y<0\\\; \; \, 1\quad y\ge 0\end{cases}$
$angle(x,y)=(1-\sigma )\pi +\sigma cos^{-1}\left (\frac{x}{r}\right)$

If however,

$\sigma=sgn(y)= \frac{y}{|y|}$

then one can let,

$\tau=\sigma^2(1-\sigma)\,$
$angle(x,y)=\tau \pi +(1-\tau ) cos^{-1}\left(\frac{x}{r}\right)$

σ and τ are similar to logic variables. These functions avoid the infinity which results from division by zero in the tangent function. The value of the angle function varies from 0 to 2π. The functions above are mathematical equivalents of the angle function found in MATHCAD.

I had to deduce these expressions myself and don't know if they are available elsewhere. --Jbergquist (talk) 21:36, 10 September 2011 (UTC)

Compare atan2. —Tamfang (talk) 06:26, 13 October 2011 (UTC)

## Circular presumption

Nearly all angles that arise are circular angles. However some are hyperbolic angles. Today the lede was restored to make this distinction. Furthermore, the restoration mentions that angles are not merely pointy things but also intervals on an arc (geometry). This issue of the ambiguity of an angle, meeting of two rays versus a measure on an arc, has been discussed above. Anon user 86.160.212.134 in London, England gives justification for the earlier reversion saying "something simpler is needed to explain the 'measure' aspect." Policy of the encyclopedia requires discussion here.Rgdboer (talk) 20:14, 3 November 2011 (UTC)

I checked your link to hyperbolic angles and there it says "In mathematics, a hyperbolic angle is a geometric figure that divides a hyperbola." My question is 'since when has an angle been a figure'.? I know hyperbolic coordinates are used and I don't think they are orthogonal but I fail completely to understand what a hyperbolic angle could be. --Damorbel (talk) 10:09, 4 November 2011 (UTC)

Please refer to details of this article: Angle here, and other articles on their talk pages.Rgdboer (talk) 01:51, 11 November 2011 (UTC)

## Angular velocity

Shouldn't there be links to angular velocity etc.? --Damorbel (talk) 11:04, 4 November 2011 (UTC)

I went ahead and added it.--RDBury (talk) 07:56, 5 November 2011 (UTC)

## Sections A formal definition and subsection Using trigonometric functions

The section A formal definition reduces to a unique subsection Using trigonometric functions. There is no formal definition there. I can not understand what is meant in the subsection which looks like a WP:OR. Thus I'll remove the section and the subsection. D.Lazard (talk) 01:23, 14 December 2011 (UTC)

## In Unicode this is symbol?

In Unicode the angle symbol ∠ is? 2220? — Preceding unsigned comment added by Reddwarf2956 (talkcontribs) 02:47, 29 August 2012 (UTC)

∠ — looks that way —Tamfang (talk) 04:52, 29 August 2012 (UTC)

## Sum of angles

The concept of the sum of two (or more) angles is used in this article without ever being formally defined. Grover cleveland (talk) 18:31, 9 October 2012 (UTC)

Well the article does say "Angle is also used to designate the measure of an angle or of a rotation". Do we need to define the sum of measures? Dbfirs 18:41, 9 October 2012 (UTC)
But the measure of an angle is not correctly defined. In fact there are various kinds of angles with different sets of possible measures, and there are as many notions of measures (in radians, in what follows):
• Non oriented angles of rays (for example rays in the space or rotations in the space): measures from 0 to π. If the sum of the measures is t>π, then the measure of the sum of the angles is 2π-t, otherwise it is t. In trigonometry, the angles are well defined by their cosine.
• Oriented angles of rays (rays in an oriented plane or rotations in the plane): measures from 0 to 2π. The sum is the sum modulo 2π. In trigonometry, one needs the sine and the cosine to define the angles.
• Non oriented angles of lines (crossing lines in the space): measures from 0 to π/2. If the sum of the measures is t>π/2, then the measure of the sum of the angles is π-t, otherwise it is t. In trigonometry, the angles are well defined by their sine, which is non negative
• Oriented angles of lines (lines in an oriented plane): measures from -π/2 to π/2 or from 0 to π. The sum is the sum modulo π. In trigonometry, the angles are well defined by their tangent.
• Oriented angle between a plane and a ray (latitude, for example): measures from -π/2 to π/2; although having the same set of measures as the preceding one, this notion of angle is different from the preceding one; it differs by {{pi]]/2 of the non oriented angle of the ray and a ray that is perpendicular to the plane.
This list is certainly not complete. IMHO, the article is confusing as it is, and should be rewritten to make a clear distinction between these different kinds of angles. D.Lazard (talk) 21:29, 9 October 2012 (UTC)
Yes, I see what you mean, but the basic measure is the same in all cases, with just a restriction on the range for particular purposes. I still think that the raw sum is clear to anyone who is not looking for problems, though it does require a bit of fudging in some circumstances. I'm happy with a re-write to clarify and improve precision of terminology, provided that it does not obscure the readability of this basic article. Mathematicians seldom agree on the exact meaning of the word "angle". Dbfirs 06:14, 10 October 2012 (UTC)
The article currently states: The size of an angle is characterized by the magnitude of the smallest rotation that maps one of the rays into the other. and also that In some contexts, ... angles that differ by an exact multiple of a full turn are effectively equivalent.
By this definition, it would be impossible to say that three angles of a square triangle add up to 270 degrees, since 270 degrees can never be "the magnitude of the smallest rotation that maps one of the rays into the other.". Grover cleveland (talk) 17:06, 12 October 2012 (UTC) (Word "triangle" removed by D.Lazard (talk) 17:51, 12 October 2012 (UTC))
I would suggest an illustrated example ABCD where angle BAC + angle CAD = angle BAD. Grover cleveland (talk) 16:59, 12 October 2012 (UTC)
IMO, a correct formulation would be ''The size of an angle (figure) is characterized .... This confirms that the notion of angle (measure) is much more complicated than it may appear and that the article says, because "angle" may measure many different things that are close but not equivalent. --D.Lazard (talk) 17:51, 12 October 2012 (UTC)
Yes, that would be a useful clarification. The problem is that the article keeps swapping between different senses of "angle". Dbfirs 08:29, 3 February 2013 (UTC)
I see that my addition of "(figure)" has been reverted, so I'll try the reverter's wording of "geometric angle" (though that's not quite a synonym to me). Dbfirs 09:02, 6 February 2013 (UTC)

## Coterminal angles and other terms

How about defining standard terms like: Standard position, initial side, terminal side, coterminal angles? Currently, coterminal angles is redirecting to a unrelated page. John W. Nicholson (talk) 06:32, 3 February 2013 (UTC)

I don't know what the standard position is. Doesn't it depend on context? (I assume that you mean either "clockwise from North" or "anticlockwise from the positive x axis".) I agree that co-terminal angles might need defining if the term were to be used in the article. I think the understanding of "initial" and "terminal" probably just needs a dictionary, or a synonym in parentheses after usage. Perhaps you could suggest what additions to the article you had in mind? Dbfirs 08:24, 3 February 2013 (UTC)

I can quote them better than I can write them. "The standard position of an angle occurs when we place its vertex at the origin of a coordinate system and its initial side on the positive x-axis as in Figure 3." Figure 3 is similar to the first one on this page: http://www.regentsprep.org/Regents/math/algtrig/ATT3/referenceAngles.htm The quote comes from a Calculus book that I have. After finding the web page I am sure that it, or something like it, can be used as a cite reference. These being defined might help with wording too. In the section Positive and negative angles, I added some changes. Please check to see if they are clear. John W. Nicholson (talk) 15:56, 3 February 2013 (UTC)

Yes, I think the author was just using the words "standard" and "position" rather than defining a well-known term, but the usage is fine. I made just one small clarification. You were correct to alter the redirection of coterminal angle. Thanks. Dbfirs 23:48, 3 February 2013 (UTC)

## Roof_pitch

Should there be a section on Roof_pitch? — Preceding unsigned comment added by Reddwarf2956 (talkcontribs) 21:15, 7 February 2013 (UTC)

I don't think so, but you could expand the short article on roof pitch with more detail of angles. Dbfirs 22:08, 7 February 2013 (UTC)

## Regionalizing equal and congruent

I see a revert back to " equal (UK) or congruent (USA) ." But, I know that equal is used here in the USA, at least to explain what congruent is. It would make sense to also use congruent in the UK also as to show that the terms are synonymous because of the term being used by other English speakers. So, the regionalization of this does not make sense. I do see that it can be explained that these terms are not used synonymously when talking about equations. John W. Nicholson (talk) 12:49, 8 February 2013 (UTC)

I had a long argument with a USA editor (see above) who claimed (and I've no reason to doubt it) that students are taught in the USA that "equal" is incorrect when used of angles. In the UK, "equal" is the standard term used in all text books. Conversely, according to the definition of congruence in our article Congruence (geometry), two angles that are equal in measure but drawn (as geometrical shapes) with different arm lengths would not be congruent in the UK sense. The regionalisation was just to use terms that are familiar to students in different regions. A reasonable compromise might be to use "equal in measure" which has the same meaning everywhere. Dbfirs 16:44, 8 February 2013 (UTC)
• Anything that would remove the regional issues is probably better with this subject. John W. Nicholson (talk) 08:28, 9 February 2013 (UTC)
I agree. Unfortunately, we cannot make US educators use "equal" or force UK educators to adopt "congruent". What do you suggest? Dbfirs 09:10, 9 February 2013 (UTC)
• I looked at what was wrote at congruent and realize there is a subtle difference in definition between congruence with a polygon and congruence with a angle which is not mentioned. The difference involves the length with the polygon while the angle does not need length. Put in another way, a line segment and a ray can have coterminal sides, so the angles are congruent, but the length of sides of the angle are not congruent. (Note the sides of angles in this http://agutie.homestead.com/files/problem/p039_geometry_help_theorem.htm in "DEFINITION 2." (just a google searched image). While we can call one side congruent in length if AB = AC, the other is described as infinite, which implies the lengths may differ.) This subtle difference leads to statements with similar polygons and congruent polygons. John W. Nicholson (talk) 10:23, 9 February 2013 (UTC)
There is no source about a linguistic regionalization here (British English vs. US English). There is no source asserting a difference between British and American mathematics. There is no reliable source asserting that using a word instead of the other is wrong. I do not know any reliable source recommending to use one word or the other. If such a recommendation exists, it could only come from an educational institution and can not apply to mathematics in the whole. Thus it does not apply to this article, which is not about teaching. If some teacher asserts that some use is wrong, this is a WP:fringe theory that has not to be reported in WP. Thus this regionalization is pure WP:OR, and I'll remove it. D.Lazard (talk) 12:42, 9 February 2013 (UTC)
Personally, I would prefer to use the unambiguous and universal "equal in measure" that is understood everywhere. I suppose that no amount of citation of texts with different usages would "prove" the regionalisation, so I'll accept your view (per policy) until I find something published on the regional difference. Dbfirs 13:37, 9 February 2013 (UTC)

## Types of angles titles

I added some titles to "Types of angles" as to manage all these angles. Feel free to change the names if you know of ones better. John W. Nicholson (talk) 17:58, 9 February 2013 (UTC)

## Turn is period?

Reddwarf2956 has been making some improvements to this article (which I support), but we seem to disagree on the definition of a turn. I don't understand how it can be a period. Am I missing something? Dbfirs 07:29, 15 September 2013 (UTC)

A circular movement is periodic, and the turn is its period. Nevertheless this concept seems too technical for this elementary article, and it is better to avoid to mention "period" here. However, I agree that this paragraph is not satisfactory, as giving a circular :-) definition. Therefore, I suggest "the turn is the angle for which the above circular arc is the complete circle". D.Lazard (talk) 09:53, 15 September 2013 (UTC)
I agree that there might be some loose association of a turn with a period in some usage, but I still don't see how a turn can be a unit of time. I agree with your suggested improvement. Dbfirs 06:25, 18 September 2013 (UTC)

## Transversal and polygons both have interior angles and exterior angles

Comments at the article edit: "I don't see what the association is for polygons. No mention at the linked article." See Angle#Polygon_related_angles. Let's add to the current statements and move them to in a different section of angle. But, we still need some references to both interior angles and exterior angles of Transversal_(geometry) with the Transversal section in angle and with polygons. Maybe combine the transversal and polygons sections together and define the terms?

John W. Nicholson (talk) 02:51, 23 September 2013 (UTC)

## Small article merger

I am trying to combine as many of the small articles on different angles into angle (per talk), but I realize that some may not fit very well. Those like transversal, polygon, and both interior angles and exterior angles don't seem to fit currently, and wiki currently has separate articles. Others like Adjacent, Complementary, Supplementary, and Vertical angles seem to easily fit. Please take look at these and decide if any more parts should be moved to this article, angle, as to keep the information or start a AfD for each. Feel free to start a AfD if you feel that it's good. — Preceding unsigned comment added by Reddwarf2956 (talkcontribs) 21:25, 24 September 2013 (UTC)