What formula/prove is there to say a triangle =180 degrees?

Question

Does anyone have a formula or written evidence that proves a triangle has a total of 180 degrees?
Thanks

Answer 1

Hi, Naz.

I think only Steiner (the name of a great geometer!) is close to answering your question properly.

The correct answer is that ther IS no proof because it's not true! If we ASSUME that the sum of the angles in a triangle is 180deg THEN we arrive at the geometry that most people are familiar with that we learn at school (called Euclidean Geometry). However, Euclidean Geometry is not the only geometry out there.

Imagine drawing a huge triangle on the surface of the earth. Start at the North Pole, trailing a huge pen, and head South until you reach the equator. Then, go around one quarter of the way round the equator (either way). Then, head due North until you reach the North Pole again. You have just drawn a triangle with three right angles! In Circular Geometry, the sum of angles in a triangle is MORE THAN 180deg!!

Hope this answers your question

Answer 2

This is true only in Euclidean geometry. It's based on the
parallel postulate that only one line can be drawn
through a point parallel to a given line.
The proof runs as follows: In the triangle below,
Call the vertices(in clockwise order) A,B,C.
▲
Now draw a line through A parallel to BC and call
the angles from right to left 1, A and 2.
Then the sum of angles 1, A, and 2 is 180 degrees.
But if 2 parallel lines are cut by a transversal,
the alternate interiour angles are congruent.
So, angle 1 = angle B and angle 2 = angle C
and thus the sum of angles A, B and C is 180 degrees.
Incidentally, this is false on a curved surface.
Astronomers have done angle measurements
on a triangle with A on the moon, B on Mars
and C on the earth and found that the sum
of the angles of such a triangle is slightly
less than 180 degrees!
Sorry I couldn't reproduce the figure for you
used in the proof.

Answer 3

Draw a triangle.

Extend all the lines beyond the corners.

Now imagine walking round the triangle. You'll finish up where you started, and you will have turned through 360 degrees.

Look at the turns you made. Each of them is next to one of the internal angles of the triangle, and is equal to 180 minus that angle. So if we add all 6 six together (inside and outside), we will have 3 times 180 = 540 degrees. But we know that the 3 on the outside total 360, so we can take that away and we are left with 180.

Draw the diagram and that should make sense.

Answer 4

I think Perspykashus is the closest.

This is to do with axioms of geometry. Similarly, 1+1=2 has no proof, it is taken to be true. If you define 1+1=3, an entirely different (but perfectly valid) type of mathematics emerges. The study of numbers and axioms is facinating.

You have been brought up with the angles in a triangle being equal to 180degrees (or pi radians), so it will be hard to accept anything else as being true.

IMPORTANTLY, this is not just a matter of the unit of degrees or how it is defined, I am actually talking about the definition of geometry. Euclidean geometry is built on the sum of the angles equalling 180 degrees, but using THE SAME MAGNITUDE of a degree, we could define the sum of the angles in a triangle to be 100, and the whole of mathematics as you know it will change.

It is a very complicated subject. It's like trying to prove that the circumference of a circle is 2pi*r. You could define the circumference of a circle to be the same as the radius. You can't picture it, it makes no sence to you because you have been raised with 2pi*r being the circumference, but it is a perfectly valid answer using a different axiom.

Go to wikipedia and search for Axioms.

Answer 5

One of the first theorems we ever learned at school gives the proof, though I don't know haw easy it is to understand without a diagram!
Imagine a triangle ABC, where A is at the top and B and C are in a straight line at the bottom.
Extend line BC, finishing at a point we will call D.
Then, at point C, draw a line parallel to AB. Call this line CE.

Angle ACE = BAC because they are alternate angles between two parallel lines.
Angle ECD = angle ABC because they are corresponding angles.

The three angles at point C which are on a straight line add up to 180°
They are the same as the angles in the triangle so the angles in the triangle add up to 180°

If you are doing this for your own curiosity why not cut out a triangle and stick the three angles together and you will see for yourself that it's true! The three angles of the triangle do make a straight line! You can do it with any size or shape of triangle

Answer 6

There is a formula which I can't remember but I do recall a trick answer put to me by my physics teacher in the 1960s. He said does a triangle always have internal angles = 180degrees? If you join 2 points on the equator to the pole then both points can originate at 90 degrees to the equator but meet at an additional angle (ie sum > 180degrees). So watch out. Make sure you are talking about a 2-D triangle.

Answer 7

Draw a triangle ABC. Let AB be the base.
Now draw a straight line through B parallel to the base.
On the straight line just drawn pick a point to the left of C and name it D. Pick another point on the line to the right of C and name it E.

Now the angle DCA = angle A and the angle ECD = angle B because they are alternate interior angles.

angle DCA + angle ACD + angle BCD = 180 degrees, because angle DCB is a straight angle which is 180 degrees.

But angle ACD is equal to angle C

So angle A + angle B + angle C = 180 degrees.

Answer 8

a + b +c = 180

Answer 9

sum of angles of an n-sided polygon is (n-2)*180

triangle (3-2)*180 = 180
rectangle = (4-2)*180 = 360
Pentagon = (5-2)*180 = 540

Answer 10

A degree is a specific measurement devised by man.
Circle is devided into 360 degrees but can also be divided into 100 grads or 2pi radians. We have decided to use degrees,it is easier.