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.
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 0 —a straight line. How many degrees is a triangle? Which brings us to the main question for today: Why is it that the interior angles of a triangle always add up to 0? As it turns out, you can figure this out by thinking about the interior and exterior angles of a triangle.
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. Now make a copy of this triangle, rotate it around 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. With me so far? As we know, if we add up the interior and exterior angles of one corner of a triangle, we always get 0. And our little drawing shows that the exterior angle in question is equal to the sum of the other two angles in the triangle.
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 0 angle.
Try making a few drawings starting with different triangles of your choosing to see this for yourself. Our lovely and elegant little drawing proves that this must be so. Or does it? Do all triangles equal degrees? Can triangles have more? Might there be some limitation to our drawing that is blinding us to some other more exotic possibility?
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. Now blow up the balloon and take a look at your triangle. What happened to it? If you have that protractor, try once again to sum up its interior angles. What happened to this sum? Do you still get 0? 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 0 as we seem to have found? Thankfully, I have the answer.
Head on over to next week's article where we started exploring the strange and wonderful world known as non-Euclidean geometry. 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. Jump to Navigation. Why Does a Triangle Have Degrees? December 10, We are currently experiencing playback issues on Safari.
If you would like to listen to the audio, please use Google Chrome or Firefox. The Quick And Dirty A triangle's angles add up to degrees because one exterior angle is equal to the sum of the other two angles in the triangle. Is this an important question? Yes, because it leads to an understanding that there are different geometries based on different axioms or 'rules of the game of geometry'.
Is it a meaningful question? Well no, at least not until we have agreed on the meaning of the words 'angle' and 'triangle', not until we know the rules of the game. Before we can say what a triangle is we need to agree on what we mean by points and lines. We are working on spherical geometry literally geometry on the surface of a sphere.
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. A great circle like the Equator cuts the sphere into two equal hemispheres. This geometry has obvious applications to distances between places and air-routes on the Earth. 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.
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. A lune is a part of the surface of the sphere bounded by two great circles which meet at antipodal points. We first consider the area of a lune and then introduce another great circle that splits the lune into triangles.
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.
Two great circles intersecting at antipodal points P and P' divide the sphere into 4 lunes. The areas of the lunes are proportional to their angles at P so the area of a lune with angle A is.
The sides of a triangle ABC are segments of three great circles which actually cut the surface of the sphere into eight spherical triangles. Between the two great circles through the point A there are four angles.
Rotating the sphere can you name the eight triangles and say whether any of them have the same area? Check your answers here. Then the area of triangle ABC is. The diagram shows a view looking down on the hemisphere which has the line through AC as its boundary.
The regions marked Area 1 and Area 3 are lunes with angles A and C respectively. Consider the lunes through B and B'.
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