The method of trying to prove that all triangles add up to 180 degrees only works on 2D surfaces and can be proved with axioms. On 2D surfaces, the proof can be said as rigorous proof because it can’t be falsified, it has been tested a lot and is known as shared knowledge. The other methods can be falsified as if you draw a triangle on a sphere the angles would not add up to 180 and therefore is falsified.
SHIP → DOCK
Changing one letter from the word ship and trying to get to the word dock would be easier if you knew that all words in the english language contain a vowel, and the conjecture that any intermediate word would have two vowels. The rigorous proof demonstrates the outcome that is expected through steps in a logical manner. With the axioms, it made the task easier to do as we believed in them.
The term proof applies differently to Math and Natural Sciences, evidence is used to prove a conjecture or theory and it can make something more true to people as they would believe it more.
Math that can be found in Nature:
Snowflakes, it has the perfect symmetry and all snowflakes are different. Sunflowers also follow the Fibonacci sequence and the nautilus shell also follows the Fibonacci sequence.