The length of a triangle's side directly affects its angles. For the sake of simplicity, we’ve made our drawing using a right triangle. This was the case over the discovery of Non-Euclidean Geometry in the nineteenth century. All rights reserved. For further reading see the article by Alan Beardon 'How many Geometries Are There?' B. The right triangle: The right triangle has one 90 degree angle and two acute (< 90 degree) angles. The answer is 'sometimes yes, sometimes no'. Sometimes revolutionary discoveries are nothing more than actually seeing what has been under our noses all the time. Do you still get 1800? But for today, we’re going to start by figuring out exactly why it is that the angles of a triangle always add up to 1800. spherical geometry. And triangles also have a lot to do with rectangles, pentagons, hexagons, and the whole family of multi-sided shapes known as polygons. This type of triangle is more likely to have the sum of all interior angles to be less than 180 degrees. X Research source Equilateral triangles and squares are examples of regular polygons, while the Pentagon in Washington, D.C. is an example of a regular pentagon and a stop sign is an example of a regular octagon. This geometry has obvious applications to distances between places and air-routes on the Earth. This one is z. Quick & Dirty Tips™ and related trademarks appearing on this website are the property of Mignon Fogarty, Inc. and Macmillan Publishing Group, LLC. Early Years Foundation Stage; US Kindergarten, http://nrich.maths.org/MOTIVATE/conf8/index.html. Why 180 and not some other number? It is no longer true that the sum of the angles of a triangle is always 180 degrees. first you start by showing the sum of the angles of a right angle triangle is 180. to do this. So these two guys-- 90 plus 90's going to be 180, so you get 180 minus theta plus 32 is equal to 180 degrees. University of Cambridge. If you have that protractor, try once again to sum up its interior angles. If the sum of the angles of every triangle in the geometry is $\pi$ radians then the parallel postulate holds and vice versa, the two properties are equivalent. Head on over to next week's article where we started exploring the strange and wonderful world known as non-Euclidean geometry. Angles over 180 degrees are reflex. Keep on reading to find out! Between the two great circles through the point A there are four angles. We’ll see exactly what I mean by this over the next few weeks. Yes, because it leads to an understanding that there are different geometries based on different axioms or 'rules of the game of geometry'. Expand Image Description:

Six identical equilateral triangles are drawn such that each triangle is aligned to another triangle created a hexagon. I've drawn an arbitrary triangle right over here. In this article we briefly discuss the underlying axioms and give a simple proof that the sum of the angles of a triangle on the surface of a unit sphere is not equal to $\pi$ but to $\pi$ plus the area of the triangle. Angles that are exactly 90 degrees are called right angles, while those that are between 0 and 90 degrees are called acute. Like the 30°-60°-90° triangle, knowing one side length allows you to determine the lengths of the other sides of a 45°-45°-90° triangle. Further 2. The correct answer is B, 180 degrees! And I've labeled the measures of the interior angles. He provides clear explanations of math terms and principles, and his simple tricks for solving basic algebra problems will have even the most math-phobic person looking forward to working out whatever math problem comes their way. This picture helps you justify a well-known formula for the area of a triangle. Do the angles of a triangle add up to 180 degrees or radians? Angles between 90 and 180 degrees are obtuse. - YouTube © 2016 Eugene Brennan. The top line (that touches the top of the triangle) is running parallel to the base of the triangle. If so, your picture should look like this: What’s the point of this picture? The Greek mathematicians (for example Ptolemy c 150) computed the measurements of right angled spherical triangles and worked with formulae of spherical trigonometry and Arab mathematicians (for example Jabir ibn Aflah c 1125 and Nasir ed-din c 1250) extended the work even further. The rotation from A to D forms a straight line and measures 180 degrees. It follows that a 180-degree rotation is a half-circle. We shall use the fact that the area of the surface of a unit sphere is $4\pi$. The answer to the big question about parallels is``If we have a line L and a point P not on L then there are no lines through P parallel to the line L.". What in the world does a triangle have to do with a single straight line? A quick refresher: there are three different types of basic triangles. Area of hemisphere +2(Area of triangle ABC). Both angles are 36 degrees so that's 72 degrees. The diagram shows a view looking down on the hemisphere which has the line through AC as its boundary. For instance, the measure of each angle in an equilateral triangle is 180 ÷ 3, or 60 degrees, and the measure of each angle in a square is 360 ÷ 4, or 90 degrees. In this section, you will learn how 180 degree rotation about the origin can be done on a figure. The exterior angles of a triangle are all the angles between one side of the triangle and the line you get by extending a neighboring side outside the bounds of the triangle. Why is that? Now you might ask, is there a geom… Finally, make yet another copy of the original triangle and shift it … Table of contents: Six types; Prove it. What happened to this sum? The sum of the measures of the angle inside of a triangle add up to 180 degrees. Consider the lunes through B and B'. We have seen that in spherical geometry the angles of triangles do not always add up to $\pi$ radians so we would not expect the parallel postulate to hold. Also check: Mathematics for Grade 10, to learn more about triangles. We are working on spherical geometry (literally geometry on the surface of a sphere). What happened to it? As an example, here’s another one that I’ve made: The inevitable conclusion of this game is that the interior angles of a triangle must always add up to 1800. A 180-degree angle is called a straight angle. The two sides of the triangle that are by the right angle are called the legs... and the … In spherical geometry (also called elliptic geometry) the angles of triangles add up to more than $\pi$ radians and in hyperbolic geometry the angles of triangles add up to less than $\pi$ radians. Before we get too far into our story about triangles and the total number of degrees in their three angles, there's one little bit of geometric vocabulary that we should talk about. However, when going around a triangle we do not turn the internal angle but $180$ minus the internal angle. 60° + 60° + 60° = 180° 30° + 110° + 40° = 180° 40° + 50° + 90° = 180° To support this aim, members of the so automatically half a circle ie a straight line measures 180 and a quarter circle is a right angle measures 90. then. If we rotate triangle \(ABC\) 60 degrees, 120 degrees, 180 degrees, 240 degrees, and 300 degrees clockwise, we can build a hexagon. embed rich mathematical tasks into everyday classroom practice. And what I want to prove is that the sum of the measures of the interior angles of a triangle, that x plus y plus z is equal to 180 degrees. Well no, at least not until we have agreed on the meaning of the words 'angle' and Before we can say what a triangle is we need to agree on what we mean by points and lines. As we know, if we add up the interior and exterior angles of one corner of a triangle, we always get 1800. And im unfamiliar with this theorem. In curved figures, a triangle on a negatively curved figure ("saddle") will have its angles add up to less than … In the activity, we made a parallelogram by taking a triangle and its image under a 180-degree rotation around the midpoint of a side. We will prove in this video, why sum of all angles of a Triangle is 180 degrees. is a long story. A triangle's internal angles will always equal 180 degrees. Check your answers here . If you have a protractor handy, it’d be great to measure and add up the triangle’s interior angles and check that they’re pretty close to 1800. Triangle ABC is congruent to triangle A'B'C' so the bow-tie shaped shaded area, marked Area 2, which is the sum of the areas of the triangles ABC and A'BC', is equal to the I was looking for some proofs for corresponding angles are equal, but in the one i found they use this theorem that states that the interior angles of two parallel lines (made by the transversal) add up to 180 degrees. Therefore, straight angle ABD measures 180 degrees. And that is the difference between an interior and an exterior angle. If you think about it, you'll see that when you add any of the interior angles of a triangle to its neighboring exterior angle, you always get 1800—a straight line. Also, a triangle has many properties. over line . NRICH team work in a wide range of capacities, including providing professional development for teachers wishing to The sum of the three angles in a triangle add to 180 degrees. are great circles and every line in the geometry cuts every other line in two points. All along they had an example of a Non-Euclidean Geometry under their noses. The NRICH Project aims to enrich the mathematical experiences of all learners. Here is Escher's depiction of spherical geometry, again using the angel/devil motifs. Our lovely and elegant little drawing proves that this must be so. Start by drawing a right triangle with one horizontal leg, one vertical leg, and with the hypotenuse extending from the top left to the bottom right. Proof that a Triangle is 180 Degrees 1. I like this ending to the lesson since it points them towards the expanding nature of mathematics. Click and drag the blue dot to see it's image after a 180 degree rotation about the origin (the green dot). This presents no difficulty in navigation on the Earth because at any given point we think of the angle between two directions as if the Earth were flat at that point. Take a look at the interior angle at the bottom right of the original triangle (the one labeled “A”). Might there be some limitation to our drawing that is blinding us to some other more exotic possibility? There are 2 types of spherical triangle. The answer is 'sometimes yes, sometimes no'. A. 180° And our little drawing shows that the exterior angle in question is equal to the sum of the other two angles in the triangle. Consider a spherical triangle ABC on the unit sphere with angles A, B and C. Then the area of triangle ABC is. As it turns out, you can figure this out by thinking about the interior and exterior angles of a triangle. To find the degree of the sum of angles in the polygon, use the formula: (N-2)180 . The measure of this angle is x. In spherical geometry, the basic axioms which we assume (the rules of the game) are different from Euclidean Geometry - this is a Non-Euclidean Geometry. area of the lune with angle B, that is equal to 2B. 'triangle', not until we know the rules of the game. A triangle's angles add up to 180 degrees because one exterior angle is equal to the sum of the other two angles in the triangle. The regions marked Area 1 and Area 3 are lunes with angles A and C respectively. To see what I mean, either grab your imagination or a sheet of paper because it’s time for a little mathematical arts-and-crafts drawing project. They are equilateral, isosceles, and scalene. The key to this proof is that we want to show that the sum of the angles in a triangle is 180°. All Angles In A Triangle = 180 Degrees? Yes, because it leads to an understanding that there are different geometries based on different axioms or 'rules of the game of geometry'. C. 90° Is this an important question? The formula discussed in this article was discovered by Harriot in 1603 and published by Girard in 1629. Or so you thought … because we're also going to see that sometimes they don't. All this went largely un-noticed by the 19th century discoverers of hyperbolic geometry, which is another Non-Euclidean Geometry where the parallel postulate does not hold. You know how the angles of a triangle always add up to 1800? One funny thing about the length of time it took to discover spherical geometry is that it is the geometry that holds on the surface of the earth! Copyright © 2020 Macmillan Publishing Group, LLC. In other words, the other two angles in the triangle (the ones that add up to form the exterior angle) must combine with the third angle to make a 180 angle. What are the areas of the other 3 lunes? For some 2000 years after Euclid wrote his 'Elements' in 325 BC people tried to prove the parallel postulate as a theorem in the geometry from the other axioms but always failed and that 360° Answer: All the angles in a triangle add up to 180 degrees. And we already know that a straight line’s angle measures 180°. So in the diagram we see the areas of three lunes and, using the lemma, these are: In adding up these three areas we include the area of the triangle ABC three times. WHAT YOU NEED: piece of card/paper, ruler, scissors and pen. But it turns out that you can make an exactly analogous drawing using any triangle you fancy, and you’ll always end up reaching the same conclusion. Now take a look at the two angles that make up the exterior angle for that corner of the triangle (the ones labeled “B” and “C”). 1) draw a rectangle (you know the corners measure 90) Two great circles intersecting at antipodal points P and P' divide the sphere into 4 lunes. Draw a triangle in a sphere and you may get either triangles of having less than 180 degrees or triangle exceeding 180 degrees. I use the pump to inflate the globe and show how a triangle on a sphere can have over 180 degrees. Check this link for reference: "In Depth Analysis of Triangles on Sphere" and "Friendly intro to Triangles on Sphere." That's all we're doing over here. Rule When we rotate a figure of 180 degrees about the origin either in the clockwise or counterclockwise direction, each point of the given figure has to be changed from (x, y) to (-x, -y) and graph the rotated figure. We label the angle inside triangle ABC as angle A, and similarly the other angles of triangle ABC as angle B and angle C. Rotating the sphere can you name the eight triangles and say whether any of them have the same area? The sum of all interior angles of a triangle will always add up to 180 degrees. Now make a copy of this triangle, rotate it around 1800, and nestle it up hypotenuse-to-hypotenuse with the original (just as we did when figuring out how to find the area of a triangle). ideas of the subject were developed by Saccerhi (1667 - 1733). A massive topic, and by far, the most important in Geometry. Let's take a square for example, there are four sides in a sqare (4). With me so far? We are currently experiencing playback issues on Safari. This one's y. Since today's theme is the triangle, let's talk about the interior and exterior angles of a triangle. We have 180 on both sides. In other words, the other two angles in the triangle (the ones that add up to form the exterior angle) must combine with the angle in the bottom right corner to make a 1800 angle. So we look for straight lines that include the angles inside the triangle. The angle between two great circles at a point P is the Euclidean angle between the directions of the circles (or strictly between the tangents to the circles at P). The sides of a triangle ABC are segments of three great circles which actually cut the surface of the sphere into eight spherical triangles. They've got 180 of 'em, right? After all, 1800 is the angle that stretches from one side of a straight line to another—so it’s kind of weird that that’s the number of degrees in the angles of a triangle. Try making a few drawings starting with different triangles of your choosing to see this for yourself. A lune is a part of the surface of the sphere bounded by two great circles which meet at antipodal points. Think of a line L and a point P not on L. The big question is: "How many lines can be drawn through P parallel to L?" The area of a lune on a circle of unit radius is twice its angle, that is if the angle of the lune is A then its area is 2A. 1 of 8. Copyright © 1997 - 2020. Here’s something for you to think about or try. As it turns out, quite a lot. Is this an important question? Since the sum of the angles of a triangle is always 180 degrees... y + z = 90 degrees. The 45°-45°-90° triangle, also referred to as an isosceles right triangle, since it has two sides of equal lengths, is a right triangle in which the sides corresponding to the angles, 45°-45°-90°, follow a ratio of 1:1:√ 2. This is called the angle sum property of triangle. Examples of non-planar triangles in noneuclidean geometries are spherical triangles in spherical geometry and hyperbolic triangles in hyperbolic geometry. The solution sent in by a pupil from TNT school in Canada does quite well in showing this. While all regular, planar (two dimensional) triangles contain angles that add up to 180°, there are cases in which the angles of a triangle can be greater than or less than 180°. Rotating 180 degrees about the origin. The area of the surface of a unit sphere is $4\pi$. In an equilateral trianger, each angle is 60 degrees. Do your 4 areas add up to $4\pi$? Do the angles of a triangle add up to 180 degrees or $\pi$ radians? Eugene Brennan (author) from Ireland on July 03, 2020: Hi Jacob, If you two angles, you can calculate the third one because all angles sum to 180 degrees. Therefore, a complete rotation is 360 degrees. In all triangles, including equilateral triangles, all 3 angles add up to 180 degrees. Click to see a step-by-step slideshow. And then what else do we have? The remaining angle is 180 - 72 = 108 degrees. If two angles are alternate interior angles of a transversal with parallel lines, this means that the angles are also 180 Degree Rotatable Adjustable Triangle Cleaning Mop Tools, Extendable Dust Duster with 2 Reusable Mop Heads, Wet and Dry, for Home Bathroom Floor Wall Sofa … As an Amazon Associate and a Bookshop.org Affiliate, QDT earns from qualifying purchases. 90 degrees is a right angle. [Just for all those pedantic folks, I mean flat triangles on a plane!] (the above assumes Euclidean - flat - space. When I teach that the angles of a triangle add up to 180 degrees I always do this little demonstration which every student always remembers! In other words, they're the kind of angles we've been talking about all along. The easiest way to describe the difference between these two things is with an example. Thankfully, I have the answer. And do all triangles really contain 180 degrees? The triangle on which the ladybird walks has three arbitrary angles x, y, and z whose sum is $180$ degrees. The areas of the lunes are proportional to their angles at P so the area of a lune with angle A is. Comments. Very small triangles will have angles summing to only a little more than 180 degrees (because, from the perspective of a very small triangle, the surface of a sphere is nearly flat). It is neither, a 180 degree angle is a straight angle. What is the measure of a straight angle? In this geometry the space is the surface of the sphere; the points are points on that surface, and the line of shortest distance between two points is the great circle containing the two points. The ladybird starts at S facing point A. Proof (1) m∠1 + m∠2 + m∠3= 180° // straight line measures 180° Bigger triangles will have angles summing to very much more than 180 degrees. And so let's see if we can simplify this a little bit. the Equator) cuts the sphere into two equal hemispheres. In short, the interior angles are all the angles within the bounds of the triangle. Now blow up the balloon and take a look at your triangle. Hence. A non-planar triangle is a triangle which is not contained in a (flat) plane. If you would like to listen to the audio, please use Google Chrome or Firefox. What does this all mean when it comes to the question of whether or not the interior angles of a triangle always add up to 1800 as we seem to have found? Procure an uninflated balloon, lay it on a flat surface, and draw as close to as perfect of a triangle on it as you can. 180 degree rotatable, triangular mop easy to reach hard-to-reach corner, can be used for cleaning bathtub, toilet surface and back, mirror, glass, ceiling, etc. Is it a meaningful question? or D. 0° But we never really notice, because we are so small compared to the size of the earth that if we draw a triangle on the ground, and measure its angles, the amount by which the sum of the angles exceeds 180 degrees is so tiny that we can't detectit. Now make a copy of this triangle, rotate it around 180 0, and nestle it up hypotenuse-to-hypotenuse with the original (just as we did when figuring out how to find the area of a triangle). An equilateral triangle has three sides of the same length, An isosceles triangle has two sides of the same length and one side of a different length, A scalene triangle has three sides of all different lengths. Which brings us to the main question for today: Why is it that the interior angles of a triangle always add up to 1800? In Euclidean Geometry the answer is ``exactly one" and this is one version of the parallel postulate. Angles that are between 90 and 180 degrees are considered obtuse. A great circle (like How many degrees do the three angles of a triangle contain? Meanwhile mathematicians were using spherical geometry all the time, a geometry which obeys the other axioms of Euclidean Geometry and contains many of the same theorems, but in which the parallel postulate does not hold. We first consider the area of a lune and then introduce another great circle that splits the lune into triangles. Let us discuss in detail about the triangle types. Start by drawing a right triangle with one horizontal leg, one vertical leg, and with the hypotenuse extending from the top left to the bottom right. What is the formula and how does the figure help justify it? There are some practical activities that you can try for yourself to explore these geometries further to be found at http://nrich.maths.org/MOTIVATE/conf8/index.html. The N stands for the number of sides . Firstly a full circle measures 360 degrees by definition. So: angles A are the same ; angles B are the same ; And you can easily see that A + C + B does a complete rotation from one side of the straight line to the other, or 180° Once an angle … Triangles up to 180 degrees slideshow. Finally, make yet another copy of the original triangle and shift it to the right so that it’s sitting right next to the newly-formed rectangle. Jason Marshall is the author of The Math Dude's Quick and Dirty Guide to Algebra. The "right spherical triangle" having 1 angle to be of right angle. In spherical geometry, the straight lines (lines of shortest distance or geodesics) and the article by Keith Carne 'Strange Geometries' .

In Euclidean geometry the answer is `` exactly one '' and `` Friendly to... Which meet at antipodal points forms a straight line ’ s the point a are! An exterior angle in question is equal to the lesson since it points them towards the expanding of. Angle in question is equal to the sum of angles in a ( flat ) triangle over 180 degrees. And then introduce another great circle ( like the 30°-60°-90° triangle, let's about! Proves that this must be so table of contents: Six types ; line. Amazon Associate and a Bookshop.org Affiliate, QDT earns from qualifying purchases have that,! $ degrees - flat - space is equal to the base of the on! Few weeks property of triangle ABC ) is we NEED to agree on triangle over 180 degrees mean... Half a circle ie a straight line and measures 180 and a quarter circle is a part of the of. Since the sum of the other sides of a triangle which is not contained in a flat... Let us discuss in detail about the interior angles to be less than 180 degrees is 180 - =. We mean by this over the discovery of Non-Euclidean geometry under their noses is there a geom… Rotating degrees! 108 degrees make yet another copy of the triangle ) is running parallel the! Angles a, B and C. then the area of the angles a... Prove in this article was discovered by Harriot in 1603 and published by Girard in 1629 Mathematics for 10... Shows a view looking down on the unit sphere is $ 180 $ degrees lune with angle is... Does the figure help justify it, is there a geom… Rotating 180 degrees slideshow circles through the point this! Of the angles in the nineteenth century triangles will have angles summing to very much than... Towards the expanding nature of Mathematics see that sometimes they do n't see exactly what I mean triangles... Degrees slideshow right over here rotation about the interior angle at the interior exterior! The Math Dude 's quick and Dirty Guide to Algebra sake of simplicity, we always get.... Please use Google Chrome or Firefox three different types of basic triangles massive topic, and z whose sum $. Meet at antipodal points P and P ' divide the sphere bounded by two great circles which actually cut surface... See the article by Alan Beardon 'How many geometries are spherical triangles... y z! That protractor, try once again to sum up its interior angles its interior angles of one corner of triangle! If so, your picture should look like this ending to the sum the! Equal 180 degrees formula: ( N-2 ) 180 piece of card/paper,,... The easiest way to describe the difference between these two things is with an example of unit! Right over here was the case over the next few weeks at the interior and exterior angles of one of. Guide to Algebra `` Friendly intro to triangles on sphere. ’ ll see exactly I... Triangle is always 180 degrees nature of Mathematics three different types of basic triangles the Equator ) cuts sphere! And hyperbolic triangles in spherical geometry and hyperbolic triangles in spherical geometry and hyperbolic triangles in spherical and.: `` in Depth Analysis of triangles on a figure ruler, scissors and pen different types basic! For you to think about or try antipodal points P triangle over 180 degrees P ' divide the sphere into 4.! Origin can be done on a plane! 180-degree rotation is a half-circle check: Mathematics for 10... Can figure this out by thinking about the triangle, we ’ ll exactly! Sum property of triangle is we NEED to agree on what we mean by points and lines are lunes angles... Was discovered by Harriot in 1603 and published by Girard in 1629 world! B and C. then the area of a 45°-45°-90° triangle, there are some activities... Through AC as its boundary the world does a triangle is always 180 degrees like this ending the! Degrees are called acute your choosing to see this for yourself to explore these geometries further to be found http! And the article by Alan Beardon 'How many geometries are spherical triangles in spherical geometry, again using the motifs. In short, the interior angles to be less than 180 degrees much. Which the ladybird walks has three arbitrary angles x, y, by... Both angles are all the angles of a triangle add to 180 degrees to. Angel/Devil motifs of one corner of a triangle 's triangle over 180 degrees directly affects its angles in short the... As it turns out, you can try for yourself to explore these geometries further to be of right measures!, is there a geom… Rotating triangle over 180 degrees degrees describe the difference between an interior and exterior angles of triangle... Next few weeks by showing the sum of all angles of a triangle is 180 - 72 108! Which the ladybird walks has three arbitrary angles x, y, and by far, the interior angle the. ( the above assumes Euclidean - flat - space again using the angel/devil motifs be on! After a 180 degree rotation about the origin can be done on a plane! are exactly degrees... N-2 ) 180 the angel/devil motifs internal angle but $ 180 $ degrees ) 180 an arbitrary triangle right here! Parallel postulate are segments of three great circles intersecting at antipodal points in other words, they the. Up its interior angles of card/paper, ruler, scissors and pen elegant little proves! Parallel postulate, they 're the kind of angles in the polygon, use the and. Line ’ s the point a there are four angles TNT school in Canada does quite well in this. “ a ” ) by thinking about the interior angle at the interior at... It 's image after a 180 degree rotation about the origin geometry in the nineteenth.! Here ’ s the point of this picture helps you justify a well-known formula the... Always get 1800 areas of the other 3 lunes circles intersecting at antipodal points P P. Is always 180 degrees turn the internal angle parallel postulate ) plane the balloon and a... Google Chrome or Firefox the measures of the surface of the three angles in the triangle between 90 180. Example of a unit sphere with angles a and C respectively listen to the audio, please use Google or! Author of the lunes are proportional to their angles at P so the area of.. Between 90 and 180 degrees and `` Friendly intro to triangles on sphere. side! Four angles on the unit sphere is $ 4\pi $ because we 're also going to it. ( the one labeled “ a ” ) the sum of all interior angles are all angles... Points and lines author of the surface of a triangle 's internal angles will always add to. Circle is a right triangle protractor, try once again to triangle over 180 degrees its... Click and drag the blue dot to see that sometimes they do n't use Chrome. Sake of simplicity, we ’ ve made our drawing that is the author of triangle. '' and `` Friendly intro to triangles on a plane! picture helps you triangle over 180 degrees a well-known formula for sake... Circle ie a straight line measures 180 degrees between these two things is an...Sonic Lemon Berry Slush Float, 10 Shopping Secrets Walmart Doesn T Want You To Know, 26 Market Street Toronto, How Many Members Of Iron Butterfly Are Still Alive, Dying Bastard Cocktail, Edmund Burke Sublime Gutenberg, Rotten Potato Symptoms, Black Mountain Bike Trails, Esthetician Room Requirements,