then the angle between b and c is:

The correct option is 180°.

Based on the provided image, we are given three vectors $\overrightarrow{a}$, $\overrightarrow{b}$, and $\overrightarrow{c}$ that satisfy the equation:

$$\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}=\overrightarrow{0}$$

along with their magnitudes:

$$\left|\overrightarrow{a}\right|=1,\left|\overrightarrow{b}\right|=1,\left|\overrightarrow{c}\right|=2$$

To find the angle between $\overrightarrow{b}$ and $\overrightarrow{c}$, we rewrite the vector equation to isolate $\overrightarrow{a}$ on one side:

$$\overrightarrow{a}=-(\overrightarrow{b}+\overrightarrow{c})$$

Taking the square of the magnitude on both sides:

$$|\overrightarrow{a}{|}^2=|-(\overrightarrow{b}+\overrightarrow{c}){|}^2$$

Using vector expansion properties, this simplifies to:

$$|\overrightarrow{a}{|}^2=|\overrightarrow{b}{|}^2+|\overrightarrow{c}{|}^2+2(\overrightarrow{b}\cdot \overrightarrow{c})$$

Substitute the given magnitudes $\left|\overrightarrow{a}\right|=1$, $\left|\overrightarrow{b}\right|=1$, and $\left|\overrightarrow{c}\right|=2$ into the equation:

$$1^2=1^2+2^2+2(\overrightarrow{b}\cdot \overrightarrow{c})$$

$$1=1+4+2(\overrightarrow{b}\cdot \overrightarrow{c})$$

$$1=5+2(\overrightarrow{b}\cdot \overrightarrow{c})$$

Solving for the dot product $\overrightarrow{b}\cdot \overrightarrow{c}$:

$$2(\overrightarrow{b}\cdot \overrightarrow{c})=-4$$

$$\overrightarrow{b}\cdot \overrightarrow{c}=-2$$

The dot product is also defined by the relation:

$$\overrightarrow{b}\cdot \overrightarrow{c}=\left|\overrightarrow{b}\right|\left|\overrightarrow{c}\right|cos\left(\theta \right)$$

where $\theta$ is the angle between the two vectors. Substituting the known magnitudes and the calculated dot product:

$$-2=\left(1\right)\left(2\right)cos\left(\theta \right)$$

$$-2=2cos\left(\theta \right)$$

$$cos\left(\theta \right)=-1$$

Since the angle $\theta$ between two vectors is in the range $[0,180°]$, we find:

$$\theta =180°$$

Thus, the angle between $\overrightarrow{b}$ and $\overrightarrow{c}$ is 180°.