To find the sizes of the unknown angles $a$, $b$, and $c$ in the image, we will use the properties of vertically opposite angles and the fact that angles on a straight line sum to $180^\circ$.

1. Identifying vertically opposite angles:

   • The angle labeled $37^\circ$ is vertically opposite to angle $b$.
   • Vertically opposite angles are equal, so: $b = 37^\circ$

2. Using the fact that angles on a straight line sum to $180^\circ$:

   • Angle $a$ and the $37^\circ$ angle form a linear pair. Therefore: $a + 37^\circ = 180^\circ$
   • Solving for $a$: $a = 180^\circ - 37^\circ = 143^\circ$

3. Using the fact that angles around a point sum to $360^\circ$:

   • The angles $a$, $b$, and $c$ are formed around a point. Therefore: $a + b + c = 360^\circ$
   • We already know $a = 143^\circ$ and $b = 37^\circ$. Substituting these values: $143^\circ + 37^\circ + c = 360^\circ$
   • Solving for $c$: $180^\circ + c = 360^\circ$ $c = 360^\circ - 180^\circ = 180^\circ$

So, the sizes of the unknown angles are: $a = 143^\circ, \quad b = 37^\circ, \quad c = 180^\circ$

Would you like more details or have any questions? Here are 5 additional questions you might find interesting:

1. How do you find the value of an unknown angle in a right triangle?
2. What is the relationship between corresponding angles and parallel lines?
3. How can you use the exterior angle theorem to solve problems?
4. What are the properties of supplementary and complementary angles?
5. How do you determine the angles in an equilateral triangle?

Tip: Always look for known angle relationships, such as vertically opposite angles and angles on a straight line, to simplify the process of finding unknown angles.