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

1. Problem Description

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

2. Solution Steps

Step 1: Recognize that the angle supplementary to $113^{\circ}$ is $180^{\circ} - 113^{\circ} = 67^{\circ}$.

So, the adjacent angle to the 113° angle is 67°.

Step 2: Notice that the two lines are parallel. The angle labeled $74^{\circ}$ and an angle corresponding to it formed by the lower parallel line and transversal are equal.

Step 3: The corresponding angle to $74^{\circ}$ plus $67^{\circ}$ must add up to $180^{\circ}$, because these are co-interior angles. However, we are given the angles $67^{\circ}$ and $113^{\circ}$, which means the angle we need is $180^{\circ}-113^{\circ} = 67^{\circ}$.

Step 4: Find the angle vertically opposite to $b$.

Angles $b$ and the angle $74^{\circ}$ plus $67^{\circ}$ must add up to $180^{\circ}$, because those two angles form a straight line. Therefore $b$ and the two smaller angles forming a straight line must add up to 180°.

Step 5: Calculate $b$.

$b = 180^{\circ} - (74^{\circ} + 67^{\circ})$

$b = 180^{\circ} - 141^{\circ}$

$b = 39^{\circ}$

3. Final Answer

The size of angle $b$ is $39^{\circ}$.