Exercise 9. Let $A B C D$ be a square with side length 1. Let $E$ be the point on the segment $[A C]$ such that $A E=1$. Calculate the value, in degrees, of the angle $\widehat{B E C}$.
Solution to Exercise 9

![](https://cdn.mathpix.com/cropped/2024_06_04_17def46ea4e6119381e4g-16.jpg?height=720&width=717&top_left_y=368&top_left_x=675)

First, since the segment $[A C]$ is a diagonal of the square, it is a line of symmetry for which points $B$ and $D$ are symmetric. In particular, it bisects the angle $\widehat{B A D}$ into two equal angles. We can deduce that $\widehat{B A C}=45^{\circ}$.

Since $A E=A B=1$, triangle $A B E$ is isosceles at $A$. We can then deduce that the angles $\widehat{B E A}$ and $\widehat{E B A}$ are equal. Since the sum of the angles in triangle $A E B$ is $180^{\circ}$, it follows that

$$
180^{\circ}=\widehat{E A B}+\widehat{A E B}+\widehat{A B E}=45^{\circ}+2 \widehat{A E B}
$$

so that $\widehat{A E B}=\frac{180^{\circ}-45^{\circ}}{2}=67.5^{\circ}$.

We can then deduce that $\widehat{B E C}=180^{\circ}-\widehat{B E A}=112.5^{\circ}$.

Alternative Solution $n^{\circ} 1$ We propose a solution that involves the intermediate calculation of $\widehat{E B C}$.

In the same way as the previous solution, we show that $\widehat{A B E}=\widehat{A E B}=67.5^{\circ}$.

Since the angle $\widehat{A B C}$ is a right angle, we have $\widehat{E B C}=90^{\circ}-\widehat{A B E}=22.5^{\circ}$. Since we also have $\widehat{E C B}=45^{\circ}$ and the sum of the angles in triangle $B C$ is $180^{\circ}$, we find

$$
\widehat{B E C}=180^{\circ}-\widehat{E B C}-\widehat{E C B}=112.5^{\circ}
$$

Graders' Comments The exercise was very well done, and almost all students found the correct answer. Many students thought to include a diagram, which is a very good point. However, it should be even more systematic to include a neat diagram on a geometry test. Attention is drawn to the importance of being careful when performing calculations and proofreading: a number of papers had the correct reasoning but lost points due to calculation errors. It is also important to avoid going in circles or assuming results that have not been proven: some papers obtained the correct result but implicitly assumed something they had not proven.