Angles in Three Squares
Drawing the right diagram can help you to prove a result about the angles in a line of squares.
Age
14 to 16
Challenge level
The picture below shows three squares.
Surprisingly, $c = a + b$.
Can you prove it?
Image
Once you've had a chance to think about how to prove it, click to reveal the diagram below.
Can you use the diagram to help you construct a proof?
Image
In the diagram below, can you find angles equal to $a, b$ and $c$ from the problem?
Image
What is special about triangle ABC?
Well done to Vid from Osnovna Å¡ola Loka ÄŒrnomelj in Slovenia, who sent us the following solution to the problem:
Angle $c = 45^\circ$ because it is the angle between the diagonal and the side of a square. To prove that $a+b=c$, I shall begin by adding points to the diagram to help with naming angles:
Image
Now we can find several angles:
- Firstly, $\angle CAE = a$ and $\angle BAD = b$.
- Since $DC$ is parallel to $AE$, by alternate angles, we know that $\angle DCA = \angle CAE$.
- Next, we can see that $\angle BCD = b$.
- Similarly, $\angle ABF = \angle CBG = b$.
- Since $\angle BHC$ is a right angle and the sum of angles in a triangle is $180^\circ$, then we have that $\angle CBH = 180^\circ - 90^\circ - b = 90^\circ - b$.
- Hence, we have that $\angle BCA = a + b$.
- In $\triangle ABC$, the line segments $AB$ and $BC$ have equal length, so we have that $\triangle ABC$ is isosceles. Therefore, $\angle BAC = \angle BCA = a+b$.
Therefore, adding the angles of $\triangle ABC$ together, we have: $$\begin{align}\angle ABC + \angle BAC + \angle ACB &= 180^\circ \\
((90^\circ - b) + b) + (a + b) + (a + b) &= 180^\circ \\
90^\circ + (a + b) + (a + b) &= 180^\circ\\
90^\circ + 2(a + b) &= 180^\circ\\
2(a + b) &= 180^\circ - 90^\circ\\
2(a + b) &= 90^\circ\\
a + b &= \frac{90^\circ}{2}\\
a + b &= 45^\circ\end{align}$$
Therefore, since $c = 45^\circ$, we have that $a + b = c$.
Why do this problem?
Geometry sometimes offers surprising results that can be proved very elegantly. This is a great example of how finding the right representation (in this case a diagram) allows a problem to be solved more easily.
Possible approach
Show students the image:Image
Invite them to draw it and measure angles a, b and c to verify that a+b=c, and give them some time to think about how they might prove it. There are many methods that use advanced techniques such as trigonometry or vectors, but there is also an elegant geometric method. Once they have had time to engage with the difficulty of the problem, show them the following image:
Image
"Can you find angles in the diagram that are the same size as angles a and b?"
"What can you say about the angle at B?"
"What can you say about the lengths AB and BC?"
"What can you deduce about the angles at A and C?"