We are given that $m\angle EFG = 6x^\circ$ and $m\angle D = (2x-4)^\circ$. We need to find the measure of $\angle D$.

1. Problem Description

We are given that $m\angle EFG = 6x^\circ$ and $m\angle D = (2x-4)^\circ$. We need to find the measure of $\angle D$.

2. Solution Steps

Since segments DE and DG are tangent to the circle at points E and G respectively, we know that $\angle DEF$ and $\angle DGF$ are right angles. Therefore, $m\angle DEF = 90^\circ$ and $m\angle DGF = 90^\circ$. Also, the sum of angles around point F is $360^\circ$. Since $\angle EFG = 6x^\circ$, then $\angle EFG$ is a reflex angle.

Consider quadrilateral DEFG. The sum of the angles in a quadrilateral is $360^\circ$.

So we have $m\angle D + m\angle DEF + m\angle EFG + m\angle DGF = 360^\circ$.

Substitute the given values: $(2x-4)^\circ + 90^\circ + 6x^\circ + 90^\circ = 360^\circ$.

Combine like terms: $8x + 176 = 360$.

Subtract 176 from both sides: $8x = 360 - 176$.

$8x = 184$.

Divide by 8: $x = \frac{184}{8} = 23$.

Now we can find $m\angle D = (2x-4)^\circ = (2(23) - 4)^\circ = (46-4)^\circ = 42^\circ$.

3. Final Answer

$m\angle D = 42^\circ$.