The figure shows a rectangle and its two diagonals. Which of the following statements is correct regarding the angles $x$ and $y$ indicated in the figure?
(a) $x<y$
(c) $2 x=3 y$
(e) $x=3 y$
(b) $x=y$
(d) $x=2 y$

![](https://cdn.mathpix.com/cropped/2024_05_01_30c9a294a58a6e4b8190g-011.jpg?height=212&width=383&top_left_y=311&top_left_x=1499)
The correct option is (d).

We have $B \widehat{C} A=D \widehat{A} C=y$ (alternate interior angles) and $B \widehat{D} A=C \widehat{B} D=y$ (symmetry). Let $O$ be the point of intersection of the two diagonals.

By drawing the segment $M N$ parallel to the sides $A D$ and $B C$ of the rectangle, we obtain $C \widehat{O} N=B \widehat{C} A=y$ and $N \widehat{O} D=$ $B \widehat{D} A=y$. Therefore,

$$
x=C \widehat{O} D=C \widehat{O} N+N \widehat{O} D=2 y
$$

![](https://cdn.mathpix.com/cropped/2024_05_01_30c9a294a58a6e4b8190g-056.jpg?height=240&width=391&top_left_y=1319&top_left_x=1438)