Prove that the sum of the exterior angles of a triangleis 360 degrees?

Question

My professor also put under the question, can you generalize to an arbitrary polygon? Hope there are some geometry people out there. He wants PROOF. Please help!!!!

Answer 1

Am assuming you don't need to prove that the sum of the interior angles for a triangle is 180 degrees...

An exterior angle is equal to 180 degrees less the size of the interior angle. So an exterior angle plus an interior angle is equal to 180 degrees.

For a triangle, there are three angles, so the sum of all the interior and exterior angles is 180 x 3 = 540 degrees.

The sum of all the interior angles of a triangle is 180 degrees.

So the sum of all the exterior angles is 540 - 180 = 360 degrees.

The general case for a polygon is as follows:

1. In the middle of your polygon, select any point.

2. From this point, draw a line to where each edge meets another edge.

3. At this point, the polygon has been split into triangles - the number of triangles it's split into is the number of sides of the polygon. Call this number n.

4. The interior angles will add up to n x 180 for all the triangles (180 for each triangle)

5. But this counts the point at the centre, so you need to deduct 360 to get the total degrees for just the polygon on its own.

6. So you've now got the total degrees for the internal angles on the polygon: 180n - 360 (test: if n=3, 180 x 3 - 360 = 180, so it works for a triangle)

7. The sum of the internal and external angles is 180n.

8. So the sum of just the external angles is 180n - (180n - 360)

9. Which equals 180n - 180n + 360

10. Which equals 360 degrees

Answer 2

The sum of the interiors angles is 180 degrees. there are 3 angles in any triangle and th sum of any exterior angle plus the interior angle which touches it is 180 degrees. 3 times 180 is 540 minus the 180 (sum of interiors) is 360 degrees.
you can prove it to any polygon with the fact that the sum of the interior angles of any polygon is (n-2)*180 while n is the number of sides.

Answer 3

In walking once around the perimeter of any polygon, you rotate exactly once, or 360 degrees. By its definition, the exterior angle is the angular difference between going straight ahead and turning the corner. If you are going clockwise, all corners make you turn to the right, thus all exterior angles have the same sign so their magnitudes add. Thus the sum of those angles is 360 degrees.

Answer 4

Draw a circle. Circumscribe a square (which is nothing more than a rectangle with all four sides equal). Draw a diagonal, which as we know divides the square into two equal parts. You now have two identical triangles, each of whose internal angles add to 180 degrees. What's left of the circle's 360 degrees? Right: 180. As further proof, draw a diameter through the circle and form any old triangle, one of whose legs is the diameter. Again, what remains is 180 degrees.

Answer 5

that's just not true.....the exterior angles of a triangle have to add up to ....for triangle ABC with angles abc......360 -a + 360 -b + 360 - c or 1080 - a+b+c or !080 - 180 = 900 degrees.

You are taking angle a out of a circle...not half a circle....same with b and c.....or a slice of pizza or a slice of pie.

That fool down there, "walking the perimeter is always gonna come up with 360 no matter what the shape....like..well gee! I finished the same direction that I started, right?

Answer 6

Just say you have a 60, 60, 60 triangle. Then the outside angles for each is going to be 120 (180-60). 120 for each equals 360.

If you have a 45/45/90 triangle, then the outside angle for the 45 will be 135 (180-45). The 90 will be 90 (180-90). 135+135+90=360.

Get it?

No matter what, you subtract each angle from 180 and add it and you will get 180.

Answer 7

let the triangle be ABC
exterior angle A=sum of the interior opposite angles B+C
exterior angle B=...............C+A
exterior angle C=...............A+B
sum of exterior angle A,B and C=2(sum of interior opposite angles A,B and C)
but the sum of interior opposite angles=180*
therefore the sum of the exterior angles=2*180=360*

Answer 8

it depends on what you mean by external angles of the triangle

a+b+c=pi

if you are saying that the ext angles are (pi-a),(pi-b) and (pi-c),we have sum of external angles = 3pi-a-b-c
= 3pi-pi =2pi =360 degrees

on the other hand,if you are saying that the ext angles are(2pi-a),(2pi-b) and (2pi-c),we have

the sum of external angles= 6pi-a -b-c = 6pi-pi=5pi =900degrees

please make your question more clear in the future