We are asked to find the sizes of angles $a$, $b$, and $c$ in the given diagram. The diagram shows two intersecting lines. One of the angles is given as $32^{\circ}$.

1. Problem Description

We are asked to find the sizes of angles $a$, $b$, and $c$ in the given diagram. The diagram shows two intersecting lines. One of the angles is given as $32^{\circ}$.

2. Solution Steps

First, we note that the sum of angles on a straight line is $180^{\circ}$. The angle adjacent to the $32^{\circ}$ angle on the same straight line is angle $a$. Thus,

$a + 32^{\circ} = 180^{\circ}$

$a = 180^{\circ} - 32^{\circ} = 148^{\circ}$.

Since the lines intersect, vertically opposite angles are equal. Therefore, angle $b$ is equal to the $32^{\circ}$ angle, so $b = 32^{\circ}$.

Similarly, angle $c$ is vertically opposite to angle $a$. Therefore, $c = a = 148^{\circ}$.

3. Final Answer

$a = 148^{\circ}$

$b = 32^{\circ}$

$c = 148^{\circ}$