A square with two attached regular pentagons. A side and a diagonal are extended. What’s the angle?
Scroll down for a solution to this problem.
Solution
The angle is 90°.
The crucial part of this solution is to drop a perpendicular from the common vertex A onto the side extension. From the external angle of the right pentagon we can deduce the angle 18°.
By using the square and the pentagon angles we see that the perpendicular actually extends a side of left pentagon. From the pentagon reflection symmetry one knows this side is parallel to the diagonal.
Finally, because internal angles formed by a transversal cutting two parallel lines are equal, we find that α=90°.
Poem
The mouse is double
2 little mices
Grey and nice
Where is the cat
To fight against that ?
Both are frightened
They will be catched
The cat opened his eyes
Much sun, no mice
May be in the evening
When he goes hunting
While I’m writing
With my computer mouse
