The problem asks to find the size of angle $b$ in the given diagram. The diagram shows two parallel lines intersected by a transversal. The angles $74^\circ$, $b$, $67^\circ$, and $113^\circ$ are marked in the diagram.

1. Problem Description

The problem asks to find the size of angle $b$ in the given diagram. The diagram shows two parallel lines intersected by a transversal. The angles $74^\circ$, $b$, $67^\circ$, and $113^\circ$ are marked in the diagram.

2. Solution Steps

First, we know that the two horizontal lines are parallel.

The angle that corresponds to the $74^\circ$ angle is equal to $74^\circ$. Since the angle $b$ and the $74^\circ$ are vertically opposite, they are equal.

Alternatively, the two lines intersecting the parallel lines form two sets of corresponding angles.

The angles $67^\circ$ and $113^\circ$ are supplementary, meaning that their sum is $180^\circ$. $67^\circ + 113^\circ = 180^\circ$, which is correct.

The angle adjacent to $113^\circ$ sums to $180^\circ$.

$180^\circ - 113^\circ = 67^\circ$.

The angle vertically opposite to $67^\circ$ is equal to $67^\circ$.

Thus, the angle inside the parallel lines on the lower side is $67^\circ$.

Consider the other intersection with the parallel lines.

The $74^\circ$ angle corresponds to an angle inside the parallel lines on the other side.

Thus this angle is also $74^\circ$.

Since angle $b$ is vertically opposite to the angle $74^\circ$, the angle $b$ must also be $74^\circ$.

3. Final Answer

$b = 74^\circ$