Three Equal Angles on a Line: Find This Side in 10 Seconds

Step by step

1. A line with three right angles
   Here's a straight line with three right angles on it.
2. 2, then 3 and 4. How tall is this side?
   This side is two. The line is cut into three and four. How tall is this side?
3. In China: "three equal angles on a line"
   In China, this is called three equal angles on a line. Here the three angles are right angles.
4. Side triangles: same shape
   The secret is that the two side triangles have the same shape. Let's see why.
5. Call this the green angle
   Look at this angle in the middle. Let's call it the green angle.
6. Yellow triangle: pink + green = 90°
   In the yellow triangle, the green angle and this pink angle add up to ninety.
7. On the line: ? + green = 90°
   On the line, the green angle and this angle also add up to ninety, because the right angle in the middle takes the rest.
8. Turn it and grow it: a perfect fit
   So this angle is pink too! Now copy the yellow triangle, turn it, and make it bigger. It fits the blue triangle exactly.
9. 2 → 4 means ×2, so 3 → 6
   Same shape means every side grows by the same amount. Two grows to four, so three grows to six. This side is six!
10. Same shape · 2 × 6 = 3 × 4
    That's the model. Three equal angles on a line give two triangles with the same shape. And here, two times six equals three times four.