Why Do a Triangle's Angles Add Up to 180°? (Picture Proof)

Step by step

1. Any triangle: the angles add up to 180°. Why?
   Draw any triangle you like. Its three angles always add up to one hundred eighty degrees. But why?
2. Way 1: tear off the three corners
   Way one is a paper trick. Tear off the three corners.
3. Point to point → a straight line = 180°
   Put them together, point to point. They line up along a straight line, and a straight line is one hundred eighty degrees.
4. Way 2: a proof with one extra line
   Tearing shows it, but it can't check every triangle. Way two proves it, with just one extra line.
5. A line through the top, parallel to the bottom
   Draw a line through the top corner, parallel to the bottom.
6. Angles inside a Z are equal
   Now look for the Z. This bottom angle and this top angle sit inside a Z, so they're equal.
7. Half a turn: the Z fits itself → the angles swap
   Why equal? Turn the Z half a turn around its middle. It lands right on itself, and the two angles trade places.
8. Same on the right side
   Same on the other side. This bottom angle matches this top angle.
9. All three on one straight line → 180°
   Now all three angles sit side by side on the top line. A straight line is one hundred eighty degrees, so they add up to one hundred eighty!
10. A proof like this: Euclid's Elements (about 2,300 years ago)
    A proof like this one is in Euclid's Elements, a Greek book from about twenty-three hundred years ago.
11. Any triangle: angles add up to 180°
    So in any triangle, the three angles add up to one hundred eighty degrees.
12. On a test: parallel line · two pairs of Z angles · straight line = 180°
    On a test, write the proof like this. Draw a line through the top corner, parallel to the bottom. The Z angles on the left are equal. The Z angles on the right are equal. The three angles at the top make a straight line. So the triangle's angles add up to one hundred eighty degrees.