1. Cutting off a triangle's corners to make a straight line
Draw a triangle on paper and cut it out. Tear off the three corners. Place their points together. They fit perfectly side-by-side with no gaps, forming a straight line. Since a straight line is 180 degrees, the sum of the angles is 180 degrees. This method is used in 5th grade math classes in Japan to check the rule [3].
However, this is not a proof. It only works for the specific triangle you cut. To say that every possible triangle, no matter how thin or large, follows this rule, we need a line of reasoning, which mathematicians call a proof. In Japanese schools, the usual proof comes in 8th grade and uses the properties of parallel lines [3]. The same proof was already written in an ancient Greek book more than 2,000 years ago. Its key step is drawing just one extra line.
2. Drawing one parallel line to bring the three angles together
Euclid's Elements, Book I, Proposition 32 states that the three interior angles of a triangle add up to two right angles [1]. Euclid did not write "180 degrees." He used the right angle as the unit.
Take a triangle with corners A, B, and C. Draw a line through corner C that is parallel to the side AB (Proposition 31). When a line crosses two parallel lines, the alternate angles are equal (Proposition 29). So the angle at A and the angle at B reappear next to angle C. Now three angles sit side by side at corner C: A, C, and B, forming a straight line [2]. This is the same proof taught in junior high school, even down to how the figure is drawn.
But wait. This proof assumes we can draw a parallel line and that those opposite angles are equal. Is that always true? It depends on the surface.
3. Euclid's fifth postulate on parallel lines and the 180-degree sum
Euclid started geometry with five basic rules, called postulates. The fifth one is long and different from the others. It says: if a line crosses two other lines and the inside angles on one side add up to less than two right angles, those two lines will eventually meet if extended far enough. Because it is longer and more complicated than the others, people tried for more than 2,000 years to derive it from the remaining postulates [5].
Note that the fifth postulate is not itself the statement "the angles add up to 180 degrees." They are two separate statements that become equivalent once the other postulates are assumed [5][6]. Another equivalent form is Playfair's axiom: "Through a point not on a line, exactly one parallel line can be drawn" [6].
This means 180 degrees is not a fact that holds in every possible world. It is a result that stands on the rule of parallel lines. If we change that rule, what happens?
4. A triangle on a sphere whose angles add up to 270 degrees
Think back to the globe. Pick the North Pole and two points on the equator whose longitudes differ by 90 degrees. Connect them. All three corners are right angles (90 degrees), so the sum is 270 degrees. This is 90 degrees more than 180, and this triangle covers 1/8 of the sphere's surface. This is a calculation from the properties of a sphere, not a figure printed in the sources.
On a sphere, a triangle is made from arcs of "great circles," which are circles whose centers are the center of the sphere. The sum of the angles of such a triangle is more than 180 degrees and less than 540 degrees, and the amount above 180 degrees (the spherical excess) is proportional to the triangle's area [7].
On a surface that curves the opposite way (hyperbolic geometry, where the parallel rule is replaced so that infinitely many parallels can pass through a point), the angles of a triangle always add up to less than 180 degrees. The missing amount is called the "angular defect," and the area is proportional to it [8]. So how the surface curves decides whether the sum is more than, exactly, or less than 180 degrees [9].
This is not just a strange imaginary world. Geometries that stay consistent even when the parallel postulate is denied were studied by Saccheri (1733) and Lambert (1766), and published by Lobachevsky in 1829–30 and Bolyai in 1832 [9]. Gauss is said to have written similar ideas in private letters but did not publish them [6].
5. Gauss's triangle of three mountain tops and the curve of space
In the 1820s, Gauss surveyed the Kingdom of Hanover. He measured a large triangle connecting three mountain tops: Hohehagen, Brocken, and Inselberg. His goals were to check the measurements of 26 smaller triangles and to study deviations in the Earth's shape [10].
There is a famous story that this survey was an experiment to test whether space is curved. But the survey can be explained without that idea [10]. Another source also calls the story a myth, as far as is known [11].
Still, as we saw with the globe, the sum of angles reflects how curved a surface is. Even if the story is a myth, thinking about what to measure to test this is a useful exercise.
6. Testing the 180-degree sum with paper triangles and a globe
You can test this with paper and a globe. First, draw three triangles of different shapes on paper, cut off the corners, and line them up. Check whether each one makes a straight line. Be careful with scissors. Remember that this only checks triangles on a flat surface.
If you have a globe, choose two meridians from the North Pole to the equator whose longitudes differ by 90 degrees. The three angles look like 90 degrees each, so the sum is 270 degrees. You can compare this with the property of a sphere that the sum is more than 180 and less than 540 degrees [7].
For further reading, the Clay Mathematics Institute's Elements Book I, Proposition 32 and its Heath translation are good primary sources [1][2]. In Japanese, the entries on the parallel postulate and on triangles are a good starting point [6][4].