In the figure, \(BCDE\) is a square, and \(ABE\) is an equilateral triangle. Prove that \(\angle AED = \angle ABC\).
Proof:
Since \(ABE\) is an triangle, all sides and all angles of \triangle ABE are equal.
That is, \(AB = BE = \) \(\longrightarrow (1)\)
Also, \(\angle AEB = \angle EBA = \angle\) \(\longrightarrow (2)\)
We also know that is a square, and in a square all sides and angles are equal.
That is, \(DE = EB = BC = CD \longrightarrow (3)\)
Similarly, \(\angle DEB = \angle \) \(= \angle BCD = \angle CDE \longrightarrow (4)\)
Now, let us try to prove that \(\angle AED = \angle ABC\).
For that matter, we should consider the triangles \(AED\) and \(ABC\).
\(\angle AED = \angle AEB + \angle BED\)
\(= \angle ABE + \angle EBC\) [From \((2)\) and \((4)\)]
\(= \angle ABC\)
Therefore, it is proved that \(\angle AED = \angle ABC\).
For that matter, we should consider the triangles \(AED\) and \(ABC\).
\(\angle AED = \angle AEB + \angle BED\)
\(= \angle ABE + \angle EBC\) [From \((2)\) and \((4)\)]
\(= \angle ABC\)
Therefore, it is proved that \(\angle AED = \angle ABC\).