Why do a triangle's angles add up to 180°?
Tear the three corners off any flat triangle and lay them side by side: they always fit exactly along a straight line, and a straight line is half a turn, 180°.
Why tho? Three levels deeper
Why is a straight line 180°?
A full turn is defined as 360°. A straight line turns you to face exactly the opposite way, which is half a turn: 180°.
How do mathematicians prove it?
Draw a line through one corner parallel to the opposite side. The two angles it makes with the triangle's sides equal the other two corners (alternate angles), and together with the third corner they sit on a straight line.
Is it true on any surface?
Only on a flat one. On a sphere, a triangle's angles add up to more than 180°: a triangle from the North Pole down to the equator and a quarter of the way around has three right angles, 270°.
Try it with paper
Cut any triangle out of paper, mark its three corners, tear them off and push them together point to point. Long, thin, wide or tiny: they always make a straight edge.
What it lets you do
Know two angles and you know the third. That's how surveyors, navigators and game engines work out angles they can't measure directly.
A nice consequence
An equilateral triangle has three equal angles, so each is 180 ÷ 3 = 60°.
Check yourself
Two angles of a triangle are 50° and 60°. What is the third?
Answer
70°, because 180 − 50 − 60 = 70.