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\begin{document}

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\section*{two pentagons solution}

\hypertarget{two_pentagons}{}\section*{{Two Pentagons}}\label{two_pentagons}

[[TwoPentagons.png:pic]]

\begin{quote}%
Two regular pentagons. What's the angle?


\end{quote}
\hypertarget{solution_by_transformation_and_similar_triangles}{}\subsection*{{Solution by [[Transformation]] and [[Similar Triangles]]}}\label{solution_by_transformation_and_similar_triangles}

[[TwoPentagonsLabelled.png:pic]]

With the points labelled as above, rotation about point $E$ of angle $72^\circ$ followed by a uniform scaling by the golden ratio also from $E$ takes $A$ to $B$ and $C$ to $D$. Therefore, applying this transformation to triangle $A E C$ produces triangle $B E D$ and so these triangles are [[similar]]. This means that $B D$ is obtained from $A C$ by rotation of $72^\circ$ (and scaling) and so angle $A \hat{F} B$ is $72^\circ$. The requested angle is therefore $108^\circ$.

\hypertarget{solution_by_invariance_principle}{}\subsection*{{Solution by [[Invariance Principle]]}}\label{solution_by_invariance_principle}

[[TwoPentagonsInvariance.png:pic]]

The above diagram contains three special cases of the diagram. The yellow pentagon is common to all three. The pink and green each share an edge with the yellow, while the orange is such that at the common vertex then the sides of it and the yellow form straight lines. Using the [[pentagon|angles in a regular pentagon]], the required angle can be seen to be $108^\circ$.



\end{document}