5. Two equilateral triangles $C E F$ and $D I H$ are positioned as shown in the diagram. The diagram indicates the measures of some angles. Find the measure of angle $x$. Provide your answer in degrees.

![](https://cdn.mathpix.com/cropped/2024_05_06_a31a204212bf5660d6dcg-2.jpg?height=577&width=966&top_left_y=1553&top_left_x=545)
Answer: 40.

Solution. Let's find the degree measures of the angles of quadrilateral $C D M E$:

![](https://cdn.mathpix.com/cropped/2024_05_06_a31a204212bf5660d6dcg-2.jpg?height=545&width=757&top_left_y=2309&top_left_x=655)

- $\angle C E M=60^{\circ}$: this is the angle of the equilateral triangle $C E F$;
- $\angle E C D=180^{\circ}-75^{\circ}=105^{\circ}$: angle $\angle E C D$ is adjacent to the angle of $75^{\circ}$;
- $\angle C D M=180^{\circ}-60^{\circ}-65^{\circ}=55^{\circ}$: angle $\angle C D M$ is adjacent to the angle formed by angle $\angle H D I$, which is $60^{\circ}$, as it is an angle of the equilateral triangle, and an angle of $65^{\circ}$;
- $\angle E M D=360^{\circ}-60^{\circ}-105^{\circ}-55^{\circ}=140^{\circ}$: the sum of the angles of quadrilateral $C E M D$ is $360^{\circ}$.

The required angle is adjacent to angle $\angle E M D$, so it is equal to $180^{\circ}-140^{\circ}=40^{\circ}$.