Question

10 in the diagram below, point e is located inside square abcd such that △abe is equilateral, and \\(\overline{ce}\\) is drawn.
what is \\(m\angle bec\\)?

1. \\(30^\circ\\)
2. \\(60^\circ\\)
3. \\(75^\circ\\)
4. \\(90^\circ\\)

diagram of square abcd with point e inside, connected to a, b, and c

Explanation:

Step1: Define square/equilateral properties

Let side length of square $ABCD$ be $s$. So $AB=BC=s$, and $\triangle ABE$ is equilateral, so $BE=AB=s$, $\angle ABE=60^\circ$.

Step2: Calculate $\angle EBC$

Square has $\angle ABC=90^\circ$.
$\angle EBC = \angle ABC - \angle ABE = 90^\circ - 60^\circ = 30^\circ$

Step3: Identify isosceles $\triangle EBC$

Since $BE=BC=s$, $\triangle EBC$ is isosceles with $BE=BC$.

Step4: Calculate base angles of $\triangle EBC$

Sum of angles in triangle is $180^\circ$. Let $\angle BEC = \angle BCE = x$.
$x + x + 30^\circ = 180^\circ$
$2x = 180^\circ - 30^\circ = 150^\circ$
$x = \frac{150^\circ}{2} = 75^\circ$

Answer:

1. $75^\circ$