why do the angles of a quadrilateral equal 360
Angles are measured in degrees, written `. The maximum angle is 360`. This is the angle all the way round a point. Half of this is the angle on a straight line, which is 180`. The video below explains how to calculate related angles, adjacent angles, interior angles and supplementary angles. Related Angles Lines AB and CD are parallel to one another (hence the on the lines). a and d are known as vertically opposite angles. Vertically opposite angles are equal. (b and c, e and h, f and g are also vertically opposite). g and c are corresponding angles. Corresponding angles are equal. (h and d, f and b, e and a are also corresponding). d and e are alternate angles. Alternate angles are equal. (c and f are also alternate). Alternate angles form a 'Z' shape and are sometimes called 'Z angles'. a and b are adjacent angles. Adjacent angles add up to 180 degrees. (d and c, c and a, d and b, f and e, e and g, h and g, h and f are also adjacent). d and f are interior angles. These add up to 180 degrees (e and c are also interior). Any two angles that add up to 180 degrees are known as supplementary angles. Angle Sum of a Triangle Using some of the above results, we can prove that the sum of the three angles inside any triangle always add up to 180 degrees. If we have a triangle, you can always draw two parallel lines like this:
Now, we know that alternate angles are equal. Therefore the two angles labelled x are equal. Also, the two angles labelled y are equal. We know that x, y and z together add up to 180 degrees, because these together is just the angle around the straight line.
So the three angles in the triangle must add up to 180 degrees. Angle Sum of a Quadrilateral A quadrilateral is a shape with 4 sides. Now that we know the sum of the angles in a triangle, we can work out the sum of the angles in a quadrilateral. For any quadrilateral, we can draw a diagonal line to divide it into two triangles. Each triangle has an angle sum of 180 degrees. Therefore the total angle sum of the quadrilateral is 360 degrees. Exterior Angles The exterior angles of a shape are the angles you get if you extend the sides. The exterior angles of a hexagon are shown: A polygon is a shape with straight sides. All of the exterior angles of a polygon add up to 360`. because if you put them all together they form the angle all the way round a point: Therefore if you have a regular polygon (in other words, where all the sides are the same length and all the angles are the same), each of the exterior angles will have size 360 the number of sides. So, for example, each of the exterior angles of a hexagon are 360/6 = 60`. Interior Angles The interior angles of a shape are the angles inside it. If you know the size of an exterior angle, you can work out the size of the interior angle next to it, because they will add up to 180` (since together they are the angle on a straight line). Exterior Angle of a Triangle The exterior angle of a triangle is equal to the sum of the interior angles at the other two vertices. In other words, x = a + b in the diagram. Proof: The angles in the triangle add up to 180 degrees. So a + b + y = 180. The angles on a straight line add up to 180 degrees.
So x + y = 180. Therefore y = 180 - x. Putting this into the first equation gives us: a + b + 180 - x = 180. Therefore a + b = x after rearranging. This is what we wanted to prove. "A quadrilateral is a closed figure which is bounded by four straight line segments. " If you start with that simple definition, you should be able to picture all kinds of things that are quadrilaterals: Quadrilaterals come in all shapes and sizes. Some of them have special names like "square," "rectangle," "parallelogram," "rhombus," and "trapezoid. " For this reading, we're going to talk very generally about quadrilaterals, so we'll start with a quadrilateral that isn't any of those "special" cases mentioned above. Did you know that inP any Pquadrilateral, the sum of the inside angles equals 360 degrees? That's not all that difficult to prove. We won'tP formally Pprove it, but I can show you a picture that'll help you to believe it: In this diagram, the quadrilateral is divided into two triangles. We don't know much about those triangles--we don't even know if they are congruent to each other. But what weP do Pknow is that the sum of their interior angles is 180 degrees. And since there are two Pof them, the total of all their angles is 360 degrees. Quadrilaterals don't just have interior angles; they also have exterior angles. The exterior angles are the angles you create by extending the sides of the quadrilateral: Now, if you're paying attention, you might think, "But that's onlyP half Pof the exterior angles! You can extend the sides in the other direction too! " Right you are!
But if you'll stop to think about it for a second (and if you remember your vertical angle theorem) you'll realize that those exterior angles are the same size as the ones drawn. So usually we just think about exterior angles going in one direction (clockwise or counter-clockwise) and remember that there are four more going in the other direction. Unless a quadrilateral is a "special" quadrilateral like a rectangle or a square, we don't know much about the exterior angles. Except we do know this: they add up to 360 degrees. How do we know this? Simple. Every interior and exterior angle forms a supplementary pair, so if you add all the interior anglesP and Pall the exterior angles, you'll get 4 straight lines, or 720 degrees. But since the sum of the interior angles is 360 degrees, that means the sum of the exterior angles is 720 - 360 = 360 degrees! "But hang on a second," you might think, "That's all well and good for a quadrilateral like the one you've drawn there, but not all quadrilaterals look the same. Some of them have a squished-in angle. " A concave quadrilateral? Like this one? "Exactly! You can't tell me that the exterior angles of that thing add up to 360 also! " Well, it turns out that, since one of the "exterior" angles is actually on theP interior, we can still make this work, as long as we agree that whenever an exterior angle is on the interior, we're going to say it has aP negative Pdegree measure. So yes, even for concave quadrilaterals, the sum of the exterior angles is 360 degrees.
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