Do These 3 Angles Make 90°? No Protractor Needed

Step by step

1. Three squares. Do the 3 angles make 90°?
   Three squares in a row. Do these three angles add up to ninety degrees? No protractor allowed.
2. Angle 1 cuts a square in half → 45°
   The first one is easy. Its line cuts a square corner in half, so it's forty-five degrees.
3. Line 2: down 1, across 2 · Line 3: down 1, across 3
   The second line goes down one square while it goes across two. The third goes down one square while it goes across three. Do those two angles make forty-five?
4. In China: the "12345 model" (1 by 2 + 1 by 3 → 45°)
   In China, this is called the one-two-three-four-five model: one by two, plus one by three, makes forty-five.
5. Way 1: a small and a big triangle share this corner
   Way one works right on this picture. Look at this small triangle and this big triangle. They share the same wide corner.
6. Shared corner = 90° + 45° = 135°
   That corner is a square corner plus half a square corner. Ninety plus forty-five is one hundred thirty-five degrees.
7. 180° − 135° = 45° → angle 2 + top angle = 45°
   A triangle's three angles add up to one hundred eighty. So in the small triangle, the other two angles make forty-five: angle two plus this top angle.
8. Flip + stretch the small one → it fits the big one
   So we only need the top angle to equal angle three. Flip the small triangle over, and stretch it a little. It fits the big triangle exactly!
9. 1 → d and d → 2 (d × d = 2 = 1 × 2)
   It fits because both sides next to the wide corner grow by the same amount. One becomes d, and d becomes two, since d times d is two, and so is one times two.
10. Top angle = angle 3 → angle 2 + angle 3 = 45°
    Same shape, same angles. The top angle lands right on angle three. So angle two plus angle three is forty-five!
11. Way 2: angle 2's line = diagonal of a 2-by-1 rectangle
    Way two is another way to see it: put the two angles side by side. Look at angle two first. Its line is the diagonal of a two-by-one rectangle.
12. Flip it → angle 2 moves to this corner
    Flip that rectangle over. The diagonal now starts at this bottom corner, and angle two comes with it. Same rectangle, same angle.
13. Angle 3: flip its 3-by-1 rectangle, then mirror it down
    Now add a second row of squares. Angle three's line is the diagonal of a three-by-one rectangle. Flip it the same way, then mirror it down.
14. Angle 2 up + angle 3 down, side by side. 45°?
    Now angle two points up, and angle three points down, from the same corner. Together they make this angle. Is it forty-five?
15. From the top end: 1 across, 2 down → bottom end
    From the end of the top line, walk one across and two down. You land right on the end of the bottom line.
16. Same 2-by-1 rectangle, lying and standing → equal, square corner
    Both lines are diagonals of the same two-by-one rectangle, one lying down and one standing up. So they're equal. Turn one a quarter turn, and you get the other, so they meet at a square corner.
17. Half a square → angle 2 + angle 3 = 45°
    Two equal sides with a square corner between them is half a square. So angle two plus angle three is forty-five degrees!
18. On a test: 135° corner · similar triangles (SAS) → angle 2 + angle 3 = 45°
    On a test, write it like this. The wide corner is one thirty-five, so angle two plus the top angle is forty-five. The sides grow evenly, so the triangles are similar. The top angle is angle three. That gives forty-five, and ninety in all.