This is a geometry problem involving angles formed by intersecting lines.

In the image, two straight lines intersect, forming four angles. You're given that one of these angles is $32^\circ$, and you need to find the sizes of the other three unknown angles labeled $a$, $b$, and $c$.

Solution Steps:

1. Angle $c$ is vertically opposite to the angle $32^\circ$. Vertically opposite angles are always equal, so: $c = 32^\circ$

2. Angle $a$ and $32^\circ$ are adjacent angles on a straight line. The sum of angles on a straight line is always $180^\circ$. Therefore: $a + 32^\circ = 180^\circ$ Solving for $a$: $a = 180^\circ - 32^\circ = 148^\circ$

3. Angle $b$ is vertically opposite to $a$, and as mentioned earlier, vertically opposite angles are equal. So: $b = 148^\circ$

Final Answers:

• $a = 148^\circ$
• $b = 148^\circ$
• $c = 32^\circ$

Would you like any additional details or clarifications?

Here are some related questions:

1. What are vertically opposite angles, and why are they always equal?
2. How do adjacent angles on a straight line add up to $180^\circ$?
3. Can we use these properties to solve for angles in polygons?
4. What happens if more lines intersect at the same point? How do we calculate the angles?
5. How can these angle properties be used in real-life situations like construction or design?

Tip: Remember that angles on a straight line always add up to $180^\circ$, and vertically opposite angles are equal. These two properties can help solve many angle problems quickly.