In the given diagram, $GI$ is a tangent to the circle at point $H$. We are given that $EF$ is parallel to $GI$. We need to find the measure of angle $EHF$. We are given that angle $FHI$ is $54^\circ$.

1. Problem Description

In the given diagram, $GI$ is a tangent to the circle at point $H$. We are given that $EF$ is parallel to $GI$. We need to find the measure of angle $EHF$. We are given that angle $FHI$ is $54^\circ$.

2. Solution Steps

Since $GI$ is tangent to the circle at $H$, and $EF$ is parallel to $GI$, we have $\angle FHI = \angle HFE = 54^\circ$ because they are alternate interior angles.

$\angle HFE = 54^\circ$.

Also, since $EF \parallel GI$, we know that $\angle EHF$ and $\angle FHI$ are supplementary angles. However, this is incorrect, and we need to consider the alternate interior angles, not supplementary angles.

We have the angle between the tangent $GI$ and the chord $HF$ is $\angle FHI = 54^\circ$. By the tangent-chord theorem, the angle between the tangent and the chord is equal to the angle subtended by the chord in the alternate segment. Therefore, $\angle FHI = \angle HGF = 54^\circ$.

However, since $EF \parallel GI$, we know that alternate interior angles are equal. Therefore, $\angle HFE = \angle FHI = 54^\circ$.

We are looking for $\angle EHF$.

Since $EF \parallel GI$, $\angle EFH = \angle FHI = 54^{\circ}$ (alternate interior angles).

Also, we know that the angle formed by tangent and chord at the point of tangency is equal to the angle in the alternate segment.

Thus, $\angle FHI = \angle HEF = 54^{\circ}$.

In triangle $EFH$, $\angle EFH = \angle HEF = 54^{\circ}$. Therefore, triangle $EFH$ is an isosceles triangle with $EH = FH$.

The sum of angles in triangle $EFH$ is $180^{\circ}$, so $\angle EHF + \angle HEF + \angle EFH = 180^{\circ}$.

Thus, $\angle EHF + 54^{\circ} + 54^{\circ} = 180^{\circ}$, which implies $\angle EHF = 180^{\circ} - 108^{\circ} = 72^{\circ}$.

3. Final Answer

The size of $\angle EHF$ is $72^{\circ}$.

Answer: B. $72^{\circ}$