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