Suppose we have a transparent circle which can be moved and placed on figures... Can you now tell which angle is bigger? Which crane was making the bigger angle?
Answer
Yes — the circle settles the argument.
Place the transparent circle so that its centre sits exactly on the vertex of the first crane's angle, and mark the points A and B where the two arms cross the circle. Now move the circle to the second crane's angle, again centre on vertex, with one arm along OA.
If the second arm OY falls outside OB, the second angle is bigger.
If it falls inside OB, the second angle is smaller.
If it lies exactly on OB, the two angles are equal.
In the picture the second crane's arm falls outside, so the second crane was making the bigger angle.
Why this works: the two angles are now compared on the same circle, so the arc cut off by the arms is a fair measure of the turn. The lengths of the beaks no longer matter at all. This is exactly the idea that leads to the protractor.
Q2.
Passing through a slit: Now, shuffle and mix up all the rotating arms. Can you identify which of the rotating arms will pass through the slit?
Answer
Only the rotating arm whose angle is exactly equal to the angle of the slit will pass through.
Situation
Will it pass?
Slit angle greater than the arms' angle
No
Slit angle less than the arms' angle
No
Slit angle equal to the arms' angle
Yes
Why it happens: the slit is a copy of one particular angle. Sliding the arms into it is the same as superimposing them, and only equal angles superimpose perfectly. Note that whether the arms pass depends only on the angle, not on the length of the straws — as long as they are shorter than the slit.
Q3.
Challenge: Reduce this angle. (The arms are cut shorter — is the angle reduced?)
Answer
No! Cutting the arms shorter does not reduce the angle — the angle is still the same.
Why it happens: the size of an angle is the amount of turn between the two arms about the vertex. Snipping the ends of the straws changes their length, but not the turn. The only way to reduce the angle is to rotate one arm closer to the other.
Check it yourself: draw an angle, measure it, then rub out the outer half of both arms and measure again. The reading does not change.