We are given a cyclic quadrilateral $ABCD$ inscribed in a circle. We are given that $\angle DBC = 47^\circ$ and $\angle ADB = 28^\circ$. We need to find the measure of the angle between the tangent to the circle at point $D$ and the side $CD$. This angle is the same as the angle $\angle DAC$.

1. Problem Description

We are given a cyclic quadrilateral $ABCD$ inscribed in a circle. We are given that $\angle DBC = 47^\circ$ and $\angle ADB = 28^\circ$. We need to find the measure of the angle between the tangent to the circle at point $D$ and the side $CD$. This angle is the same as the angle $\angle DAC$.

2. Solution Steps

Since $ABCD$ is a cyclic quadrilateral, we know that $\angle DAC = \angle DBC$.

Also, $\angle ACB = \angle ADB$ because they subtend the same arc $AB$.

Given $\angle DBC = 47^\circ$, we know that $\angle DAC = 47^\circ$.

Also given $\angle ADB = 28^\circ$, we know $\angle ACB = 28^\circ$.

The angle between the tangent at $D$ and the chord $CD$ is equal to the angle in the alternate segment, which is $\angle DAC$.

Therefore, the angle between the tangent at $D$ and $CD$ is $\angle DAC$, which is equal to $\angle DBC$.

3. Final Answer

The measure of the angle between the tangent to the circle at point $D$ and the side $CD$ is $47^\circ$.