In plane geometry, an angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle.^{[1]} Angles formed by two rays lie in a plane, but this plane does not have to be a Euclidean plane. Angles are also formed by the intersection of two planes in Euclidean and other spaces. These are called dihedral angles. Angles formed by the intersection of two curves in a plane are defined as the angle determined by the tangent rays at the point of intersection. Similar statements hold in space, for example, the spherical angle formed by two great circles on a sphere is the dihedral angle between the planes determined by the great circles.
Angle is also used to designate the measure of an angle or of a rotation. This measure is the ratio of the length of a circular arc to its radius. In the case of a geometric angle, the arc is centered at the vertex and delimited by the sides. In the case of a rotation, the arc is centered at the center of the rotation and delimited by any other point and its image by the rotation.
The word angle comes from the Latin word angulus, meaning "corner"; cognate words are the Greek ἀγκύλος (ankylοs), meaning "crooked, curved," and the English word "ankle". Both are connected with the ProtoIndoEuropean root *ank, meaning "to bend" or "bow".^{[2]}
Euclid defines a plane angle as the inclination to each other, in a plane, of two lines which meet each other, and do not lie straight with respect to each other. According to Proclus an angle must be either a quality or a quantity, or a relationship. The first concept was used by Eudemus, who regarded an angle as a deviation from a straight line; the second by Carpus of Antioch, who regarded it as the interval or space between the intersecting lines; Euclid adopted the third concept, although his definitions of right, acute, and obtuse angles are certainly quantitative.^{[3]}
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Transcription
A couple of videos ago we introduced the idea of the length of a vector. That equals the length. And this was a neat idea because we're used to the length of things in two or threedimensional space, but it becomes very abstract when we get to n dimensions. If this has a hundred components, at least for me, it's hard to visualize a hundred dimension vector. But we've actually defined it's notion of length. And we saw that this is actually a scalar value. It's just a number. In this video, I want to attempt to define the notion of an angle between vectors. As you can see, we're building up this mathematics of vectors from the ground up, and we can't just say, oh, I know what an angle is because everything we know about angles and even lengths, it just applies to what we associate with two or threedimensional space. But the whole study of linear algebra is abstracting these ideas into multidimensional space. And I haven't even defined what dimension is yet, but I think you understand that idea to some degree already. When people talk about one or two or three dimensions. So let's say that I have some vector let's say I have two vectors, vectors a and b. They're nonzero and they're members of Rn. And I don't have a notion of the angle between them yet, but let me just draw them out. Let me just draw them as if I could draw them in two dimensions. So that would be vector a right there. Maybe that's vector b right there. And then this vector right there would be the vector a minus b. And you can verify that just the way we've learned to add and subtract vectors. Or you know, this is heads to tails. So b plus a minus b is of course, going to be vector a. And that all just works out there. To help us define this notion of angle, let me construct another triangle that's going to look a lot like this one. But remember, I'm just doing this for our simple minds to imagine it in two dimensions. But these aren't necessarily twodimensional beasts. These each could have a hundred components. But let me make another triangle. Well, it should look similar. Say it looks like that. And I'm going to define the sides of the triangles to be the lengths of each of these vectors. Remember, the lengths of each of these vectors, I don't care how many components there are, they're just going to be your numbers. So the length of this side right here is just going to be the length of a. The length of this side right here is just going to be the length of vector a minus vector b. And the length of this side right here is going to be the length of vector b. Now the first thing we want to make sure is that we can always construct a triangle like that. And so under what circumstances could we not construct a triangle like this? Well we wouldn't be able construct a triangle like this if this side. if b, if the magnitude so let me write this down. It's kind of a subtle point, but I want to make this very clear. In order to define an angle, I want to be comfortable that I can always make this construction. And I need to make sure that let me write reasons why I couldn't make this construction. Well what if the magnitude of b was greater than, or the length of vector b was greater than the length of vector a plus the length of vector a minus b? In two dimensions, I could never draw a triangle like that then because you would have this length plus this length would be shorter than this thing right here. So you could never construct it. And I could do with all the sides. What if this length was larger than one of these two sides? Or what if that length was larger than one of those two sides? I could just never draw a twodimensional triangle that way. So what I'm going to do is I'm going to use the triangle the vector triangle inequality to prove that each of these sides is less than or equal to the sum of the other sides. I could do the same thing. Let me make the point clear. I could show that if a, for whatever reason, was greater than the other sides plus b, then I wouldn't be able to create a triangle. And the last one of course is if a minus b, for whatever reason, was greater than the other two sides, I just wouldn't be able to draw a triangle in a plus b. So I need to show that for any vectors, any real vectors nonzero, real vectors that are members of Rn that none of these can ever happen. I need to prove that none of those can happen. So what does the triangle inequality tell us? The triangle inequality tells us that if I have the sum of two vectors, if I take the length of the sum of two vectors, that that is always going to be less than and these are nonzero vectors. This is always going to be less than or equal to the sum of each of their individual lengths. So let's see if we can apply that to this triangle right here. So what is the magnitude, the length of a? Well I can rewrite vector a. What is vector a equal to? Vector a is equal to vector b plus vector a minus b. I mean I'm just rewriting the vector here. I'm just rewriting a here as a sum of the other two vectors. Nothing fancy there. I haven't used the triangle inequality or anything. I've just used my definition of vector addition. But here now, if I put little parentheses here, now I can apply the triangle inequality. And I say, well, you know what? This is going to be, by the triangle inequality, which we've proved, it's going to be less than or equal to the lengths of each of these vectors. Vector b plus the length of vector a minus b. So we know that the length of a is less than the sum of that one and that one. So we don't have to worry about this being our problem. We know that that is not true. Now let's look at b. So is there any way that I can rewrite b as a sum of two other vectors? Well sure. I can write it as a sum of a plus, let me put it this way. If that vector right there is a minus b, the same vector in the reverse direction is going to be the vector b minus a. So a plus the vector b minus a. That's the same thing as b. And you can see it right here. The a's would cancel out and you're just left with the b there. Now by the triangle inequality, we know that this is less than or equal to the length of vector a plus the length of vector b minus a. Now you're saying hey, Sal, you're dealing with b minus a. This is the length of a minus b. And I can leave this for you to prove it based on our definition of vector lengths, but the length of b minus a is equal to minus 1 times a minus b. And I'll leave it to you to say that look, these lengths are equal. Because essentially I could leave that, but I think you can take that based on just the visual depiction of them that they're the exact same vectors, just in different directions. And I have to be careful with length because it's not just in two dimensions. But I think you get the idea and I'll leave that for you to prove that these lengths are the same thing. So we know that b is less than the length of those two things. So we don't have to worry about that one right there. Finally, a minus b. The magnitude or the length of vector a minus b. Well I can write that as the length of or I can write that as vector a plus vector minus b. If we just put a minus b right there and go in the other directions, we could say minus b, which would be in that direction plus a would give us our vector a minus b. Actually, I don't even have to go there. That's obvious from this. I just kind of put the negative in the parentheses. Well the triangle inequality, and this might seem a little mundane to you, but it really shows us that we can always define a regular planar triangle based on these vectors in this way. It tells us this is less than or equal to the length of our vector a plus the length of minus b. And I just said and you could prove it to yourself, that this is the same thing as the length of b. So we just saw that this is definitely less than those two. This is definitely less than those two. And that is definitely less than those two. None of the reasons that would keep us from constructing a triangle are valid. So we can always construct a triangle in this way from any arbitrary nonzero vectors in Rn. We can always construct this. Now, to define an angle, let me redraw it down here. Let me redraw the vectors, maybe a little bit bigger. That's vector a. This is vector b. And then let me just draw it this way. This is the vector right there. That is the vector a minus b. And we said we're going to define a corresponding regular, run of the mill, vanilla triangle whose lengths are defined by the lengths of the vectors, by the vector lengths. So this is the length of b, that side. This is the length of a minus b. And then this is the length of a. Now that I know that I can always construct a triangle like this, I can attempt to define or actually, I will define my definition of an angle between two vectors. So we know what an angle means in this context. This is just a regular, run of the mill, geometric triangle. Now, my definition of an angle between two vectors I'm going to say so this is what I'm trying to define. This is what I'm going to define. These can have arbitrary number of components, so it's hard to visualize. But I'm going to define this angle as the corresponding angle in a regular, run of the mill triangle where the sides of the run of the mill triangle are the two vectors and then the opposite side is the subtraction, is the length of the difference between the two vectors. This is just the definition. I'm defining this, the angle between two vectors in Rn that could have an arbitrary number of components, I'm defining this angle to be the same as this angle, the angle between the two sides, the two lengths of those vectors in just a regular, run of the mill triangle. Now, what can I do with this? Well, can we find a relationship between all of these things right here? Well sure. If you remember from your trigonometry class, and if you don't, I've proved it in the playlist. You have the law of cosines. And I'll do it with an arbitrary triangle right here just because I don't want to confuse you. So if this is side a, b, and c and this is theta, the law of cosines tells us that c squared is equal to a squared plus b squared minus 2ab cosine of theta. I always think of it as kind of a broader Pythagorean theorem because this thing does not have to be a right angle. It accounts for all angles. If this becomes a right angle, then this term disappears and you're just left with the Pythagorean theorem. But we've proven this. This applies to just regular, run of the mill triangles. And lucky for us, we have a regular, run of the mill triangle here. So let's apply the law of cosines to this triangle right here. And the way I drew it, they correspond. The length of this side squared. So that means the length of a minus b squared. Length of vector a minus vector b, that's just the length of that side. So I'm just squaring that side. It equals the length of vector b squared plus the length of vector a squared minus 2 times the length of I'll just write two times length of vector a times the length of vector b times the cosine of this angle right here. Times the cosine of that angle. And I'm defining this angle between these two vectors to be the same as this angle right there. So if we know this angle, by definition, we know that angle right there. Well, we know that the square of our lengths of a vector when we use our factor definition of length, that this is just the same thing as a vector dotted with itself. So that's a minus b dot a minus b. It's all going to be equal to this whole stuff on the righthand side. But let me simplify the lefthand side of this equation. a minus b dot a minus b, this is the same thing as a dot a those two terms minus a dot b. And then I have minus b dot a. Those two terms right there. And then you have the minus b dot minus b. That's the same thing as a plus b dot b. Remember, this is just a simplification of the lefthand side. And I can rewrite this. a dot a, we know that's just the length of a squared. a dot b and b dot a are the same thing, so we have two of these. So this right here, this term right there will simplify to minus 2 times a dot b. And then finally, b dot b. We know that that's just the length of b squared. I just simplified or maybe I just expanded that's a better word. When you go from one term here to three terms, you can't say you simplified it. But I expanded just the lefthand side and so this has to be equal to the righthand side by the law of cosines. So that is equal to I almost feel like instead of rewriting it, let me just copy and paste it. What did I just do? Copy, edit. Copy and paste. There you go. I don't know it that was worth it. But maybe I saved a little bit of time. So that is equal to that right there. And then we can simplify. We have a length of a squared here, length of a squared there. Subtract it from both sides. The length of b squared here, length of b squared there. Subtract it from both sides. And then, what can we do? We can divide both sides by minus 2 because everything else has disappeared. And so that term and that term will both become 1's. And all we're left with is the vector a dot the vector b. And this is interesting because all of a sudden we're getting a relationship between the dot products of two vectors. We've kind of gone away from their definition by lengths. But the dot product of two vectors is equal to the product of their lengths, their vector lengths. And they can have an arbitrary number of components. Times the cosine of the angle between them. Remember, this theta, I said this is the same as when you draw this kind of analogous, regular triangle. But I'm defining the angle between them to be the same as that. So I can say that this is the angle between them. And obviously, the idea of between two vectors, it's hard to visualize if you go beyond three dimensions. But now we have it at least, mathematically defined. So if you give me two vectors we can now, using this formula that we've proved using this definition up here, we can now calculate the angle between any two vectors using this right here. And just to make it clear, what happens if a is a and maybe it's not clear from that definition, so I'll make it clearer here that by definition, if a is equal to some scalar multiple of b where c is greater than 0, we'll define theta to be equal to 0. And if c is less than 0, so a is collinear, but goes in the exact opposite direction, we'll define theta to be equal to 180 degrees. And that's consistent with what we understand about just twodimensional vectors. If they're collinear and kind of the scalar multiples the same. That means a looks something like that and b looks something like that. So we say oh, that's a 0 angle. And if they go the other way, if a looks something like this is the case where a is just going in the other direction from b. a goes like that and b goes like that, we define the angle between them to be 180 degrees. But everything else is pretty well defined by the triangle example. I had to make the special case of these because it's not clear you really get a triangle in these cases because the triangle kind of disappears. It flattens out if a and b are on top of each other or if they're going in the exact opposite direction. So that's why I wanted to make a little bit of a side note right there. Now, using this definition of the angle between the vectors, we can now define the idea of perpendicular vectors. So we can now say perpendicular vectors this is another definition and this won't be earth shattering, but it kind of is because we've generalized this to vectors that have an arbitrary number of components. We're defining perpendicular to mean the theta between two vectors a and b are perpendicular if the angle between them is 90 degrees. And we can define that. We can take two vectors, dot them. Take their dot product. Figure out their two lengths and then you could figure out the angle between them. And if it's 90 degrees. you can say that they are perpendicular angles. And I want to be very clear here that this is actually not defined for the 0 vector right here. So this situation right here, not defined for the 0 vector. Because if you have the 0 vector, then this quantity right here is going to be 0 and then this quantity right here is going to be 0. And there's no clear definition for your angle. If this is 0 right here, you did 0 is equal to 0 times cosine of theta. And so if you wanted to solve for theta you'd get cosine of theta is equal to 0/0, which is undefined. But what we can do is create a slightly more general word than the word perpendicular. So you have to have a defined angle to even talk about perpendicular. If the angle between two vectors is 90 degrees, we're saying by definition, those two vectors are perpendicular. But what if we made the statement and we can if you look at them, if the angle between two vectors is 90 degrees, what does that mean? So let's say that theta is 90 degrees. Let me draw a line here. Let's say that theta is 90 degrees. Theta is equal to 90 degrees. What does this formula tell us? It tells us that a dot b is equal to the length of a times the length of b times cosine of 90 degrees. What's cosine of 90 degrees? It's 0. You can review your unit circle if that doesn't make a lot of sense. But that is equal to 0, so this whole term is going to be equal to 0. So if theta is equal to 90 degrees, then a dot b is equal to 0. And so this is another interesting takeaway. If a and b are perpendicular, then their dot product is going to be equal to 0. Now if their dot product is equal to 0, can we necessarily say that they are perpendicular? Well what if a or b is the 0 vector? The 0 vector let me call it z for 0 vector. Or I could just draw. The 0 vector dot anything is always going to be equal to 0. So does that mean that the 0 vector is perpendicular to everything? Well no. Because the 0 vector I said, we have to have the notion of an angle between things in order to use the word perpendicular. So we can't use the 0 vector. We can't say just because two vectors dot products are equal to 0 that they are perpendicular. And that's because the 0 vector would mess that up because the 0 vector is not defined. But if we say, and we have been saying, that a and b are nonzero, if they are nonzero vectors, then we can say that if a and b are nonzero and their dot product is equal to 0, then a and b are perpendicular. So now it goes both ways. But what if we just have this condition right here? What if we just have the condition that a dot b is equal to 0? It seems like that's kind of just a simple, pure condition. And we can write a word for that. And these words are often used synonymously, but hopefully you understand the distinction now. We can say that if two vectors dot product is equal to 0, we will call them orthogonal. As I always say, spelling isn't my best subject. But this is kind of a neat idea. This tells us that well, that all perpendicular vectors are orthogonal. And it also tells us that the 0 vector is orthogonal to everything else. To everything, even to itself. The 0 dot 0 vector you still get 0. So by definition, it's orthogonal. So for the first time probably in your mathematical career, you're seeing that the words you know, every time you first got exposed to the words perpendicular and orthogonal in geometry or maybe in physics or wherever else, they were always kind of the same words. But now I'm introducing a nice, little distinction here and you can kind of be a little smart aleck with teachers. Oh, you know, it's perpendicular only is the vectors aren't if neither of them are 0 vector. Otherwise, if their dot product is 0, you can only say that they're orthogonal. But if they're nonzero you could say that they're orthogonal and perpendicular. But anyway, I thought that I would introduce this little distinction for you in case you have someone that likes to trip you up with words. But it also I think highlights that we are building a mathematics from the ground up and we have to be careful about the words we use. And we have to be very precise about our definitions. Because if we're not precise about our definitions and we build up a bunch of mathematics on top of this and do a bunch of proofs, one day we might scratch our heads and read some type of weird ambiguity. And it might have all came out of the fact that we weren't precise enough in defining what some of these terms mean. Well anyway, hopefully you found this useful. We can now take the angle or we can now determine the angle between vectors with an arbitrary number of components.
Contents
Identifying angles
In mathematical expressions, it is common to use Greek letters (α, β, γ, θ, φ, . . . ) to serve as variables standing for the size of some angle. (To avoid confusion with its other meaning, the symbol π is typically not used for this purpose.) Lower case Roman letters (a, b, c, . . . ) are also used, as are upper case Roman letters in the context of polygons. See the figures in this article for examples.
In geometric figures, angles may also be identified by the labels attached to the three points that define them. For example, the angle at vertex A enclosed by the rays AB and AC (i.e. the lines from point A to point B and point A to point C) is denoted ∠BAC (in Unicode U+2220 ∠ ANGLE) or . Sometimes, where there is no risk of confusion, the angle may be referred to simply by its vertex ("angle A").
Potentially, an angle denoted, say, ∠BAC might refer to any of four angles: the clockwise angle from B to C, the anticlockwise angle from B to C, the clockwise angle from C to B, or the anticlockwise angle from C to B, where the direction in which the angle is measured determines its sign (see Positive and negative angles). However, in many geometrical situations it is obvious from context that the positive angle less than or equal to 180 degrees is meant, and no ambiguity arises. Otherwise, a convention may be adopted so that ∠BAC always refers to the anticlockwise (positive) angle from B to C, and ∠CAB to the anticlockwise (positive) angle from C to B.
Types of angles
Individual angles
 An angle equal to 0° or not turned is called a zero angle.
 Angles smaller than a right angle (less than 90°) are called acute angles ("acute" meaning "sharp").
 An angle equal to 1/4 turn (90° or π/2 radians) is called a right angle. Two lines that form a right angle are said to be normal, orthogonal, or perpendicular.
 Angles larger than a right angle and smaller than a straight angle (between 90° and 180°) are called obtuse angles ("obtuse" meaning "blunt").
 An angle equal to 1/2 turn (180° or π radians) is called a straight angle.
 Angles larger than a straight angle but less than 1 turn (between 180° and 360°) are called reflex angles.
 An angle equal to 1 turn (360° or 2π radians) is called a full angle, complete angle, round angle or a perigon.
 Angles that are not right angles or a multiple of a right angle are called oblique angles.
The names, intervals, and measured units are shown in a table below:
Name  zero  acute  right angle  obtuse  straight  reflex  perigon  
Units  Interval  

Turns  0  (0, 1/4)  1/4  (1/4, 1/2)  1/2  (1/2, 1)  1  
Radians  0  (0, 1/2π)  1/2π  (1/2π, π)  π  (π, 2π)  2π  
Degrees  0°  (0, 90)°  90°  (90, 180)°  180°  (180, 360)°  360°  
Gons  0^{g}  (0, 100)^{g}  100^{g}  (100, 200)^{g}  200^{g}  (200, 400)^{g}  400^{g} 
Equivalence angle pairs
 Angles that have the same measure (i.e. the same magnitude) are said to be equal or congruent. An angle is defined by its measure and is not dependent upon the lengths of the sides of the angle (e.g. all right angles are equal in measure).
 Two angles which share terminal sides, but differ in size by an integer multiple of a turn, are called coterminal angles.
 A reference angle is the acute version of any angle determined by repeatedly subtracting or adding straight angle (1/2 turn, 180°, or π radians), to the results as necessary, until the magnitude of result is an acute angle, a value between 0 and 1/4 turn, 90°, or π/2 radians. For example, an angle of 30 degrees has a reference angle of 30 degrees, and an angle of 150 degrees also has a reference angle of 30 degrees (180–150). An angle of 750 degrees has a reference angle of 30 degrees (750–720).^{[4]}
Vertical and adjacent angle pairs
When two straight lines intersect at a point, four angles are formed. Pairwise these angles are named according to their location relative to each other.
 A pair of angles opposite each other, formed by two intersecting straight lines that form an "X"like shape, are called vertical angles or opposite angles or vertically opposite angles. They are abbreviated as vert. opp. ∠s.^{[5]}
 The equality of vertically opposite angles is called the vertical angle theorem. Eudemus of Rhodes attributed the proof to Thales of Miletus.^{[6]}^{[7]} The proposition showed that since both of a pair of vertical angles are supplementary to both of the adjacent angles, the vertical angles are equal in measure. According to a historical Note,^{[7]} when Thales visited Egypt, he observed that whenever the Egyptians drew two intersecting lines, they would measure the vertical angles to make sure that they were equal. Thales concluded that one could prove that all vertical angles are equal if one accepted some general notions such as: all straight angles are equal, equals added to equals are equal, and equals subtracted from equals are equal.
 In the figure, assume the measure of Angle A = x. When two adjacent angles form a straight line, they are supplementary. Therefore, the measure of Angle C = 180 − x. Similarly, the measure of Angle D = 180 − x. Both Angle C and Angle D have measures equal to 180 − x and are congruent. Since Angle B is supplementary to both Angles C and D, either of these angle measures may be used to determine the measure of Angle B. Using the measure of either Angle C or Angle D we find the measure of Angle B = 180 − (180 − x) = 180 − 180 + x = x. Therefore, both Angle A and Angle B have measures equal to x and are equal in measure.
 Adjacent angles, often abbreviated as adj. ∠s, are angles that share a common vertex and edge but do not share any interior points. In other words, they are angles that are side by side, or adjacent, sharing an "arm". Adjacent angles which sum to a right angle, straight angle or full angle are special and are respectively called complementary, supplementary and explementary angles (see "Combine angle pairs" below).
A transversal is a line that intersects a pair of (often parallel) lines and is associated with alternate interior angles, corresponding angles, interior angles, and exterior angles.^{[8]}
Combining angle pairs
There are three special angle pairs which involve the summation of angles:
 Complementary angles are angle pairs whose measures sum to one right angle (1/4 turn, 90°, or π/2 radians). If the two complementary angles are adjacent their nonshared sides form a right angle. In Euclidean geometry, the two acute angles in a right triangle are complementary, because the sum of internal angles of a triangle is 180 degrees, and the right angle itself accounts for ninety degrees.
 The adjective complementary is from Latin complementum, associated with the verb complere, "to fill up". An acute angle is "filled up" by its complement to form a right angle.
 The difference between an angle and a right angle is termed the complement of the angle.^{[9]}
 If angles A and B are complementary, the following relationships hold:
 (The tangent of an angle equals the cotangent of its complement and its secant equals the cosecant of its complement.)
 The prefix "co" in the names of some trigonometric ratios refers to the word "complementary".
 Two angles that sum to a straight angle (1/2 turn, 180°, or π radians) are called supplementary angles.
 If the two supplementary angles are adjacent (i.e. have a common vertex and share just one side), their nonshared sides form a straight line. Such angles are called a linear pair of angles.^{[10]} However, supplementary angles do not have to be on the same line, and can be separated in space. For example, adjacent angles of a parallelogram are supplementary, and opposite angles of a cyclic quadrilateral (one whose vertices all fall on a single circle) are supplementary.
 If a point P is exterior to a circle with center O, and if the tangent lines from P touch the circle at points T and Q, then ∠TPQ and ∠TOQ are supplementary.
 The sines of supplementary angles are equal. Their cosines and tangents (unless undefined) are equal in magnitude but have opposite signs.
 In Euclidean geometry, any sum of two angles in a triangle is supplementary to the third, because the sum of internal angles of a triangle is a straight angle.
 Two angles that sum to a complete angle (1 turn, 360°, or 2π radians) are called explementary angles or conjugate angles.
 The difference between an angle and a complete angle is termed the explement of the angle or conjugate of an angle.
 An angle that is part of a simple polygon is called an interior angle if it lies on the inside of that simple polygon. A simple concave polygon has at least one interior angle that is a reflex angle.
 In Euclidean geometry, the measures of the interior angles of a triangle add up to π radians, 180°, or 1/2 turn; the measures of the interior angles of a simple convex quadrilateral add up to 2π radians, 360°, or 1 turn. In general, the measures of the interior angles of a simple convex polygon with n sides add up to (n − 2)π radians, or 180(n − 2) degrees, (2n − 4) right angles, or (n/2 − 1) turn.
 The supplement of an interior angle is called an exterior angle, that is, an interior angle and an exterior angle form a linear pair of angles. There are two exterior angles at each vertex of the polygon, each determined by extending one of the two sides of the polygon that meet at the vertex; these two angles are vertical angles and hence are equal. An exterior angle measures the amount of rotation one has to make at a vertex to trace out the polygon.^{[11]} If the corresponding interior angle is a reflex angle, the exterior angle should be considered negative. Even in a nonsimple polygon it may be possible to define the exterior angle, but one will have to pick an orientation of the plane (or surface) to decide the sign of the exterior angle measure.
 In Euclidean geometry, the sum of the exterior angles of a simple convex polygon will be one full turn (360°). The exterior angle here could be called a supplementary exterior angle. Exterior angles are commonly used in Logo Turtle Geometry when drawing regular polygons.
 In a triangle, the bisectors of two exterior angles and the bisector of the other interior angle are concurrent (meet at a single point).^{[12]}^{:p. 149}
 In a triangle, three intersection points, each of an external angle bisector with the opposite extended side, are collinear.^{[12]}^{:p. 149}
 In a triangle, three intersection points, two of them between an interior angle bisector and the opposite side, and the third between the other exterior angle bisector and the opposite side extended, are collinear.^{[12]}^{:p. 149}
 Some authors use the name exterior angle of a simple polygon to simply mean the explement exterior angle (not supplement!) of the interior angle.^{[13]} This conflicts with the above usage.
 The angle between two planes (such as two adjacent faces of a polyhedron) is called a dihedral angle.^{[9]} It may be defined as the acute angle between two lines normal to the planes.
 The angle between a plane and an intersecting straight line is equal to ninety degrees minus the angle between the intersecting line and the line that goes through the point of intersection and is normal to the plane.
Measuring angles
The size of a geometric angle is usually characterized by the magnitude of the smallest rotation that maps one of the rays into the other. Angles that have the same size are said to be equal or congruent or equal in measure.
In some contexts, such as identifying a point on a circle or describing the orientation of an object in two dimensions relative to a reference orientation, angles that differ by an exact multiple of a full turn are effectively equivalent. In other contexts, such as identifying a point on a spiral curve or describing the cumulative rotation of an object in two dimensions relative to a reference orientation, angles that differ by a nonzero multiple of a full turn are not equivalent.
In order to measure an angle θ, a circular arc centered at the vertex of the angle is drawn, e.g. with a pair of compasses. The ratio of the length s of the arc by the radius r of the circle is the measure of the angle in radians.
The measure of the angle in another angular unit is then obtained by multiplying its measure in radians by the scaling factor k/2π, where k is the measure of a complete turn in the chosen unit (for example 360 for degrees or 400 for gradians):
The value of θ thus defined is independent of the size of the circle: if the length of the radius is changed then the arc length changes in the same proportion, so the ratio s/r is unaltered. (Proof. The formula above can be rewritten as k = θr/s. One turn, for which θ = n units, corresponds to an arc equal in length to the circle's circumference, which is 2πr, so s = 2πr. Substituting n for θ and 2πr for s in the formula, results in k = nr/2πr = n/2π.) ^{[nb 1]}
Angle addition postulate
The angle addition postulate states that if B is in the interior of angle AOC, then
The measure of the angle AOC is the sum of the measure of angle AOB and the measure of angle BOC. In this postulate it does not matter in which unit the angle is measured as long as each angle is measured in the same unit.
Units
Units used to represent angles are listed below in descending magnitude order. Of these units, the degree and the radian are by far the most commonly used. Angles expressed in radians are dimensionless for the purposes of dimensional analysis.
Most units of angular measurement are defined such that one turn (i.e. one full circle) is equal to n units, for some whole number n. The two exceptions are the radian and the diameter part.
 Turn (n = 1)
 The turn, also cycle, full circle, revolution, and rotation, is complete circular movement or measure (as to return to the same point) with circle or ellipse. A turn is abbreviated τ, cyc, rev, or rot depending on the application, but in the acronym rpm (revolutions per minute), just r is used. A turn of n units is obtained by setting k = 1/2π in the formula above. The equivalence of 1 turn is 360°, 2π rad, 400 grad, and 4 right angles. The symbol τ can also be used as a mathematical constant to represent 2π radians. Used in this way (k = τ/2π) allows for radians to be expressed as a fraction of a turn. For example, half a turn is τ/2 = π.
 Quadrant (n = 4)
 The quadrant is 1/4 of a turn, i.e. a right angle. It is the unit used in Euclid's Elements. 1 quad. = 90° = π/2 rad = 1/4 turn = 100 grad. In German the symbol ^{∟} has been used to denote a quadrant.
 Sextant (n = 6)
 The sextant (angle of the equilateral triangle) is 1/6 of a turn. It was the unit used by the Babylonians,^{[15]}^{[16]} and is especially easy to construct with ruler and compasses. The degree, minute of arc and second of arc are sexagesimal subunits of the Babylonian unit. 1 Babylonian unit = 60° = π/3 rad ≈ 1.047197551 rad.
 Radian (n = 2π = 6.283 . . . )
 The radian is the angle subtended by an arc of a circle that has the same length as the circle's radius. The case of radian for the formula given earlier, a radian of n = 2π units is obtained by setting k = 2π/2π = 1. One turn is 2π radians, and one radian is 180/π degrees, or about 57.2958 degrees. The radian is abbreviated rad, though this symbol is often omitted in mathematical texts, where radians are assumed unless specified otherwise. When radians are used angles are considered as dimensionless. The radian is used in virtually all mathematical work beyond simple practical geometry, due, for example, to the pleasing and "natural" properties that the trigonometric functions display when their arguments are in radians. The radian is the (derived) unit of angular measurement in the SI system.
 Clock position (n = 12)
 A clock position is the relative direction of an object described using the analogy of a 12hour clock. One imagines a clock face lying either upright or flat in front of oneself, and identifies the twelve hour markings with the directions in which they point.
 Hour angle (n = 24)
 The astronomical hour angle is 1/24 of a turn. As this system is amenable to measuring objects that cycle once per day (such as the relative position of stars), the sexagesimal subunits are called minute of time and second of time. These are distinct from, and 15 times larger than, minutes and seconds of arc. 1 hour = 15° = π/12 rad = 1/6 quad. = 1/24 turn = 16+2/3 grad.
 (Compass) point or wind (n = 32)
 The point, used in navigation, is 1/32 of a turn. 1 point = 1/8 of a right angle = 11.25° = 12.5 grad. Each point is subdivided in four quarterpoints so that 1 turn equals 128 quarterpoints.
 Hexacontade (n = 60)
 The hexacontade is a unit of 6° that Eratosthenes used, so that a whole turn was divided into 60 units.
 Pechus (n = 144–180)
 The pechus was a Babylonian unit equal to about 2° or 2+1/2°.
 Binary degree (n = 256)
 The binary degree, also known as the binary radian (or brad), is 1/256 of a turn.^{[17]} The binary degree is used in computing so that an angle can be efficiently represented in a single byte (albeit to limited precision). Other measures of angle used in computing may be based on dividing one whole turn into 2^{n} equal parts for other values of n.^{[18]}
 Degree (n = 360)
 The degree, denoted by a small superscript circle (°), is 1/360 of a turn, so one turn is 360°. The case of degrees for the formula given earlier, a degree of n = 360° units is obtained by setting k = 360°/2π. One advantage of this old sexagesimal subunit is that many angles common in simple geometry are measured as a whole number of degrees. Fractions of a degree may be written in normal decimal notation (e.g. 3.5° for three and a half degrees), but the "minute" and "second" sexagesimal subunits of the "degreeminutesecond" system are also in use, especially for geographical coordinates and in astronomy and ballistics.
 Diameter part (n = 376.99 . . . )
 The diameter part (occasionally used in Islamic mathematics) is 1/60 radian. One "diameter part" is approximately 0.95493°. There are about 376.991 diameter parts per turn.
 Grad (n = 400)
 The grad, also called grade, gradian, or gon, is 1/400 of a turn, so a right angle is 100 grads. It is a decimal subunit of the quadrant. A kilometre was historically defined as a centigrad of arc along a great circle of the Earth, so the kilometer is the decimal analog to the sexagesimal nautical mile. The grad is used mostly in triangulation.
 Milliradian
 The milliradian (mil or mrad) is defined as a thousandth of a radian, which means that a rotation of one turn consists of 2000π mil (or approximately 6283.185... mil), and almost all scope sights for firearms are calibrated to this definition. In addition there are three other derived definitions used for artillery and navigation which are approximately equal to a milliradian. Under these three other definitions one turn makes up for exactly 6000, 6300 or 6400 mils, which equals spanning the range from 0.05625 to 0.06 degrees (3.375 to 3.6 minutes). In comparison, the true milliradian is approximately 0.05729578... degrees (3.43775... minutes). One "NATO mil" is defined as 1/6400 of a circle. Just like with the true milliradian, each of the other definitions exploits the mil's handby property of subtensions, i.e. that the value of one milliradian approximately equals the angle subtended by a width of 1 meter as seen from 1 km away (2π/6400 = 0.0009817... ≈ 1/1000).
 Minute of arc (n = 21,600)
 The minute of arc (or MOA, arcminute, or just minute) is 1/60 of a degree = 1/21,600 turn. It is denoted by a single prime ( ′ ). For example, 3° 30′ is equal to 3 × 60 + 30 = 210 minutes or 3 + 30/60 = 3.5 degrees. A mixed format with decimal fractions is also sometimes used, e.g. 3° 5.72′ = 3 + 5.72/60 degrees. A nautical mile was historically defined as a minute of arc along a great circle of the Earth.
 Second of arc (n = 1,296,000)
 The second of arc (or arcsecond, or just second) is 1/60 of a minute of arc and 1/3600 of a degree. It is denoted by a double prime ( ″ ). For example, 3° 7′ 30″ is equal to 3 + 7/60 + 30/3600 degrees, or 3.125 degrees.
 Milliarcsecond (n = 1,296,000,000)
 mas
 Microarcsecond (n = 1,296,000,000,000)
 µas
Positive and negative angles
Although the definition of the measurement of an angle does not support the concept of a negative angle, it is frequently useful to impose a convention that allows positive and negative angular values to represent orientations and/or rotations in opposite directions relative to some reference.
In a twodimensional Cartesian coordinate system, an angle is typically defined by its two sides, with its vertex at the origin. The initial side is on the positive xaxis, while the other side or terminal side is defined by the measure from the initial side in radians, degrees, or turns. With positive angles representing rotations toward the positive yaxis and negative angles representing rotations toward the negative yaxis. When Cartesian coordinates are represented by standard position, defined by the xaxis rightward and the yaxis upward, positive rotations are anticlockwise and negative rotations are clockwise.
In many contexts, an angle of −θ is effectively equivalent to an angle of "one full turn minus θ". For example, an orientation represented as −45° is effectively equivalent to an orientation represented as 360° − 45° or 315°. Although the final position is the same, a physical rotation (movement) of −45° is not the same as a rotation of 315° (for example, the rotation of a person holding a broom resting on a dusty floor would leave visually different traces of swept regions on the floor).
In threedimensional geometry, "clockwise" and "anticlockwise" have no absolute meaning, so the direction of positive and negative angles must be defined relative to some reference, which is typically a vector passing through the angle's vertex and perpendicular to the plane in which the rays of the angle lie.
In navigation, bearings or azimuth are measured relative to north. By convention, viewed from above, bearing angles are positive clockwise, so a bearing of 45° corresponds to a northeast orientation. Negative bearings are not used in navigation, so a northwest orientation corresponds to a bearing of 315°.
Alternative ways of measuring the size of an angle
There are several alternatives to measuring the size of an angle by the angle of rotation. The grade of a slope, or gradient is equal to the tangent of the angle, or sometimes (rarely) the sine. A gradient is often expressed as a percentage. For very small values (less than 5%), the grade of a slope is approximately the measure of the angle in radians.
In rational geometry the spread between two lines is defined as the square of the sine of the angle between the lines. As the sine of an angle and the sine of its supplementary angle are the same, any angle of rotation that maps one of the lines into the other leads to the same value for the spread between the lines.
Astronomical approximations
Astronomers measure angular separation of objects in degrees from their point of observation.
 0.5° is approximately the width of the sun or moon.
 1° is approximately the width of a little finger at arm's length.
 10° is approximately the width of a closed fist at arm's length.
 20° is approximately the width of a handspan at arm's length.
These measurements clearly depend on the individual subject, and the above should be treated as rough rule of thumb approximations only.
Angles between curves
The angle between a line and a curve (mixed angle) or between two intersecting curves (curvilinear angle) is defined to be the angle between the tangents at the point of intersection. Various names (now rarely, if ever, used) have been given to particular cases:—amphicyrtic (Gr. ἀμφί, on both sides, κυρτός, convex) or cissoidal (Gr. κισσός, ivy), biconvex; xystroidal or sistroidal (Gr. ξυστρίς, a tool for scraping), concavoconvex; amphicoelic (Gr. κοίλη, a hollow) or angulus lunularis, biconcave.^{[19]}
Bisecting and trisecting angles
The ancient Greek mathematicians knew how to bisect an angle (divide it into two angles of equal measure) using only a compass and straightedge, but could only trisect certain angles. In 1837 Pierre Wantzel showed that for most angles this construction cannot be performed.
Dot product and generalisations
In the Euclidean space, the angle θ between two Euclidean vectors u and v is related to their dot product and their lengths by the formula
This formula supplies an easy method to find the angle between two planes (or curved surfaces) from their normal vectors and between skew lines from their vector equations.
Inner product
To define angles in an abstract real inner product space, we replace the Euclidean dot product ( · ) by the inner product , i.e.
In a complex inner product space, the expression for the cosine above may give nonreal values, so it is replaced with
or, more commonly, using the absolute value, with
The latter definition ignores the direction of the vectors and thus describes the angle between onedimensional subspaces and spanned by the vectors and correspondingly.
Angles between subspaces
The definition of the angle between onedimensional subspaces and given by
in a Hilbert space can be extended to subspaces of any finite dimensions. Given two subspaces , with , this leads to a definition of angles called canonical or principal angles between subspaces.
Angles in Riemannian geometry
In Riemannian geometry, the metric tensor is used to define the angle between two tangents. Where U and V are tangent vectors and g_{ij} are the components of the metric tensor G,
Hyperbolic angle
A hyperbolic angle is an argument of a hyperbolic function just as the circular angle is the argument of a circular function. The comparison can be visualized as the size of the openings of a hyperbolic sector and a circular sector since the areas of these sectors correspond to the angle magnitudes in each case. Unlike the circular angle, the hyperbolic angle is unbounded. When the circular and hyperbolic functions are viewed as infinite series in their angle argument, the circular ones are just alternating series forms of the hyperbolic functions. This weaving of the two types of angle and function was explained by Leonhard Euler in Introduction to the Analysis of the Infinite.
Angles in geography and astronomy
In geography, the location of any point on the Earth can be identified using a geographic coordinate system. This system specifies the latitude and longitude of any location in terms of angles subtended at the centre of the Earth, using the equator and (usually) the Greenwich meridian as references.
In astronomy, a given point on the celestial sphere (that is, the apparent position of an astronomical object) can be identified using any of several astronomical coordinate systems, where the references vary according to the particular system. Astronomers measure the angular separation of two stars by imagining two lines through the centre of the Earth, each intersecting one of the stars. The angle between those lines can be measured, and is the angular separation between the two stars.
In both geography and astronomy, a sighting direction can be specified in terms of a vertical angle such as altitude /elevation with respect to the horizon as well as the azimuth with respect to north.
Astronomers also measure the apparent size of objects as an angular diameter. For example, the full moon has an angular diameter of approximately 0.5°, when viewed from Earth. One could say, "The Moon's diameter subtends an angle of half a degree." The smallangle formula can be used to convert such an angular measurement into a distance/size ratio.
See also
 Angle bisector
 Angular velocity
 Argument (complex analysis)
 Astrological aspect
 Central angle
 Clock angle problem
 Dihedral angle
 Exterior angle theorem
 Great circle distance
 Inscribed angle
 Irrational angle
 Phase angle
 Protractor
 Solid angle for a concept of angle in three dimensions.
 Spherical angle
 Transcendent angle
 Trisection
 Zenith angle
Notes
 ^ This approach requires however an additional proof that the measure of the angle does not change with changing radius r, in addition to the issue of "measurement units chosen." A smoother approach is to measure the angle by the length of the corresponding unit circle arc. Here "unit" can be chosen to be dimensionless in the sense that it is the real number 1 associated with the unit segment on the real line. See Radoslav M. Dimitrić for instance.^{[14]}
References
 ^ Sidorov 2001
 ^ Slocum 2007
 ^ Chisholm 1911; Heiberg 1908, pp. 177–178
 ^ "Mathwords: Reference Angle". www.mathwords.com. Archived from the original on 23 October 2017. Retrieved 26 April 2018.
 ^ Wong & Wong 2009, pp. 161–163
 ^ Euclid. The Elements. Proposition I:13.
 ^ ^{a} ^{b} Shute, Shirk & Porter 1960, pp. 25–27.
 ^ Jacobs 1974, p. 255.
 ^ ^{a} ^{b} Chisholm 1911
 ^ Jacobs 1974, p. 97.
 ^ Henderson & Taimina 2005, p. 104.
 ^ ^{a} ^{b} ^{c} Johnson, Roger A. Advanced Euclidean Geometry, Dover Publications, 2007.
 ^ D. Zwillinger, ed. (1995), CRC Standard Mathematical Tables and Formulae, Boca Raton, FL: CRC Press, p. 270 as cited in Weisstein, Eric W. "Exterior Angle". MathWorld.
 ^ Dimitrić, Radoslav M. (2012). "On Angles and Angle Measurements" (PDF). The Teaching of Mathematics. XV (2): 133–140. Archived (PDF) from the original on 20190117. Retrieved 20190806.
 ^ Jeans, James Hopwood (1947). The Growth of Physical Science. p. 7.
 ^ Murnaghan, Francis Dominic (1946). Analytic Geometry. p. 2.
 ^ "ooPIC Programmer's Guide  Chapter 15: URCP". ooPIC Manual & Technical Specifications  ooPIC Compiler Ver 6.0. Savage Innovations, LLC. 2007 [1997]. Archived from the original on 20080628. Retrieved 20190805.
 ^ Hargreaves, Shawn. "Angles, integers, and modulo arithmetic". blogs.msdn.com. Archived from the original on 20190603. Retrieved 20190805.
 ^ Chisholm 1911; Heiberg 1908, p. 178
 Henderson, David W.; Taimina, Daina (2005), Experiencing Geometry / Euclidean and NonEuclidean with History (3rd ed.), Pearson Prentice Hall, p. 104, ISBN 9780131437487
 Heiberg, Johan Ludvig (1908), Heath, T. L. (ed.), Euclid, The Thirteen Books of Euclid's Elements, 1, Cambridge: Cambridge University Press.
 Sidorov, L. A. (2001) [1994], "Angle", in Hazewinkel, Michiel (ed.), Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 9781556080104
 Jacobs, Harold R. (1974), Geometry, W. H. Freeman, pp. 97, 255, ISBN 9780716704560
 Slocum, Jonathan (2007), Preliminary IndoEuropean lexicon — Pokorny PIE data, University of Texas research department: linguistics research center, retrieved 2 Feb 2010
 Shute, William G.; Shirk, William W.; Porter, George F. (1960), Plane and Solid Geometry, American Book Company, pp. 25–27
 Wong, Takwah; Wong, Mingsim (2009), "Angles in Intersecting and Parallel Lines", New Century Mathematics, 1B (1 ed.), Hong Kong: Oxford University Press, pp. 161–163, ISBN 9780198001775
Attribution
 This article incorporates text from a publication now in the public domain: Chisholm, Hugh, ed. (1911), "Angle", Encyclopædia Britannica, 2 (11th ed.), Cambridge University Press, p. 14
External links
Wikimedia Commons has media related to Angles (geometry). 
 Encyclopædia Britannica, 2 (9th ed.), 1878, pp. 29–30 ,
 Proximity construction of an angle in decimal degrees with the third intercept theorem
 Angle Bisectors in a Quadrilateral at cuttheknot
 Constructing a triangle from its angle bisectors at cuttheknot
 Various angle constructions with compass and straightedge
 Complementary Angles animated demonstration. With interactive applet
 Supplementary Angles animated demonstration. With interactive applet
 Angle definition pages with interactive applets that are also useful in a classroom setting. Math Open Reference
 Construction of an angle^{[permanent dead link]} Site geometry