Three Squares Problem

[bob012345]

TL;DR

Using only elementary geometry (not even trigonometry), prove that angle C in the figure equals the sum of angles A and B.

This problem is another from Martin Gardner’s book Mathematical Circus but originates from a fourth grade extra credit question in a Moscow school long ago. It makes me wonder how America won the Space Race! The problem statement is simply;

Using only elementary geometry (not even trigonometry), prove that angle C in the figure equals the sum of angles A and B.

There are apparently dozens of proofs of this so I suggest just showing your solution and others can come up with different solutions. I selected High School because it probably is at least that level for today’s readers. Have fun!

 

[Ibix]

Spoiler

The red triangle replicates the 3×1 triangle in the OP; the blue and green ones replicate the 2×1 triangle. These three triangles contribute one side each to the triangle $PQR$.

The angle $\angle PRQ$ is a right angle, since it is constructed from the green right angled triangle's $90-B$ angle and the blue triangle's $B$ angle. The sides $PR$ and $QR$ are manifestly equal lengths, so the triangle is right angled isoceles and hence angles $\angle QPR$ and $\angle PQR$ are 45°. From the angle $\angle QPR$, we can see that they are also equal to $A+B$, so $A+B=45°$. $C$ is also 45°, obvious from symmetry. Thus $C=A+B$.

BBC News once posted a challenging geometry problem from a Chinese maths exam taken aged 16, comparing it to the British exams taken at the same age.

A Chinese colleague observed that a Chinese child would have spent a lot of time learning to do similar puzzles and would therefore have solved it faster (using geometry) than I did (using vectors). But was the time spent learning to solve those time well spent? Did it give them maths skills, or just teach them tricks for solving geometry problems that will never come up in any practical situation?