All triangles equal to 180 degrees if you add the angles together. How is this true?

Question

Please explain why all triangles equal up to 180 degrees.

Answer 1

One way to prove this is to make any triagle out of a piece of paper. Now rip the angels off. Put the points together. You will see that the angles form a straight line. This means that the angles add up to 180 degrees.

There are other proofs that involve parallel lines and trig.

http://whyslopes.com/Euclidean-Geometry-Introduction/geom11_Triangle_Angle_Sum.html
(the above link is a proof with parallel lines)

Answer 2

For euclidean (meaning on a flat surface like paper) the simplest example is a right triangle. Draw it - note that it is half of a rectangle. Complete the rectangle and it's pretty easy to see that each triangle contributes half of the angles which would be 180 degrees.

What about triangles other than a right triangle? - try this: draw a random triangle. Now, make a line parallel to one edge and going through the point at the top. Put on sides on that go between these two lines at the other two points and are at 90 degrees (right angles). This is also a rectangle. Now draw a line from point at the top to the bottom line - again at right angles. You have now created two right triangles which together make the original.

The two right angles from these smaller triangles make 180 degrees together - the remaining, when all added up, must make 180 as well. No matter how you make a triangle, the corners simply must add up to 180 - its comes along with the constraint of having three sides that meet.

Answer 3

A triangle is defined as a three-sided plane made up of three line segments connecting three points. It is further assumed in the definition that the sum of all of the angles of a triangle are 180 degrees. This can be arrived at logically because if the sum of the angles of a triangle are greater than 180 degrees, then it would be possible to make a triangle with two right angles. However, when two right angles are formed using three line segments, it does not create creates a three-sided plane. Instead, it creates the beginnings of a rectangle (of which, a square is a subset). It is also not possible to create a polygon with less than 180 degrees because a two-sided polygon is impossible to form since a line segment can be extended indefinitely and the third angle formed will always make the sum of all angles 180 degrees.. Thus, the sum of all angles of a triangle are 180 degrees. With mathematics, it is necessary to assume that axioms are true as that is the definition of an axiom.

Answer 4

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It is important to understand the the sum of the measures of the three angles inside of a triangle always add up to 180. If you know that, you will always know what the three angles will add up to because this is true for every triangle. So consider the a triangle with measures 30, 40 and x. We need to know x, which is a variable. To find x we construct an equation and then solve for x. Since you know that the three angles will always add up to 180, you know the equation will equal 180 and that you are adding the three angles together. 30 + 40 + x = 180 ---> x = 110 Subtract 70 from each side to isolate x on one side of the equation. The unknown angle is 110 degrees. ____________________________ Practice: You have a right triangle with two angles each 45 degrees in measure. Satisfy yourself that the third angle is 90 degrees. Remember that all right triangles have one degree equal to 90 degrees. In general, with Geometry it is good to learn the general principles and then use those general principles to solve specific questions. In a lot of math you can often just mechanically work through the numbers without knowing why things work, just that they do, but with Geometry you have know generally why something works and then apply it, so concentrate on learning the axioms, definitions and so on, such as "the sum of the internal angles of any triangle is 180 degrees." With that knowledge if you are given two angles of any triangle you can find the third!

Answer 5

Do All Triangles Equal 180

Answer 6

This holds only in euclidean geometry, with its set of postulates. If you vary the fifth postulate (the one on the parallel lines) you get a new set of theorems and a different system (non euclidean). In the two non eulidean systems the angles of a triangle DON'T equal up to 180 degrees, but add up to more than 180 degrees in one system and less than 180 in the other.
So is it really true that all triangles equal to 180 degrees if you add the angles together?
Well, the geometry we study at school works very well for many practical and theoretical purposes. However in particular contexts, like in the study of the universe (space is curved), scientists find non euclidean geometry (Riemann geometry) more fit.

Answer 7

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RE:
All triangles equal to 180 degrees if you add the angles together. How is this true?
Please explain why all triangles equal up to 180 degrees.

Answer 8

A right angled triangle is half of a rectangle. And any rectangle has the sum of its angles 360 degrees. So the right angled triangle has a half of that value which is 180 degrees. All other triangles work on this basis, since if you draw a clone of any triangle and put them together, they will form a quaternion, such as a square, a rectangle, or a kite.

Answer 9

For any triangle ABC, where A, B, and C are points on a plane, the following is true. Line AB creates angles BAC and ABC. BAC and ABC deviate from the line at a combined distance of BAC+ABC degrees. Angle ACB is equal to the difference of 180 degrees ( the measurement of a straight line in degrees) minus the sum of the two angles listed.

Answer 10

Let us assume that you are given triangle ABC, lying in some euclidean plane.

The question is to show that the sum of the measurements of angles A, B, and C is equal to 180 degrees. As we are in a Euclidean plane, there is a line through point A that is parallel to line BC.

_______A__________
/\
/ \
/ \
B /___\ C


Now we have a set of parallel lines, and a couple of transversals ( AB and AC are the transversals ). After identifying congruent angles using the standard parallel lines cut by a transversal theorems, you will be finished.