First we need to convert degrees to radians by multiplying by $\displaystyle \frac{\pi}{180^{\circ}}$:

$\displaystyle 30^{\circ}\times \frac{\pi}{180^{\circ}}=\frac{\pi}{6}$

Now we can write: 

$\displaystyle \frac{x+\pi}{x-\pi}=\frac{\pi}{6}\Rightarrow 6(x+\pi)=\pi(x-\pi)$

$\displaystyle \Rightarrow 6x+6\pi=x\pi-\pi^2\Rightarrow 6x-x\pi=-6\pi-\pi^2$

$\displaystyle \Rightarrow x(6-\pi)=-\pi(\pi+6)$

$\displaystyle \Rightarrow x=\frac{-\pi(\pi+6)}{6-\pi}=\frac{\pi(\pi+6)}{\pi-6}$

First, we need to convert $\displaystyle 72^{\circ}$ to radians by multiplying by $\displaystyle \frac{\pi}{180^{\circ}}$: 

$\displaystyle 72^{\circ}\times \frac{\pi}{180^{\circ}}=\frac{2\pi}{5}$

Now we can solve the following equation for $\displaystyle x$:

$\displaystyle \frac{72^{\circ}-\pi}{2x-\pi}=3\Rightarrow \frac{\frac{2\pi}{5}-\pi}{2x-\pi}=3$

$\displaystyle \Rightarrow {\frac{2\pi}{5}-\pi}=3(2x-\pi)\Rightarrow \frac{2\pi-5\pi}{5}=6x-3\pi$

$\displaystyle \Rightarrow -3\pi=30x-15\pi\Rightarrow 30x=12\pi$

$\displaystyle \Rightarrow x=\frac{12\pi}{30}\Rightarrow x=\frac{2\pi}{5}$

Since $\displaystyle 180^\circ=\pi\ \text{radians}$, we can solve by setting up a proportion:

$\displaystyle \frac{x}{\frac{5\pi}{6}}=\frac{180^\circ}{\pi}$

Cross multiply and solve.

$\displaystyle 180^\circ*\frac{5\pi}{6}=x*\pi$

$\displaystyle \frac{900\pi}{6}=x*\pi$

$\displaystyle 150\pi=x*\pi$

$\displaystyle \frac{150\pi}{\pi}=x$

$\displaystyle 150^\circ=x$

First, we need to simplify the expression:

$\displaystyle \frac{\frac{\pi}{3}-\frac{\pi}{6}}{\frac{1}{3}+\frac{1}{6}}=\frac{\frac{2\pi-\pi}{6}}{\frac{2+1}{6}}=\frac{\pi}{3}$

Now multiply by $\displaystyle \frac{180^{\circ}}{\pi}$:

$\displaystyle \frac{\pi}{3}\times \frac{180^{\circ}}{\pi}=\frac{180^{\circ}}{3}=60^{\circ}$

First, we need to simplify the expression:

$\displaystyle \frac{45^{\circ}+30^{\circ}}{45-30}=\frac{75^{\circ}}{15}=5^{\circ}$

First, we need to simplify the expression:

$\displaystyle \frac{\frac{2\pi}{7}+\frac{\pi}{14}}{\frac{1}{7}}=\frac{\frac{4\pi+\pi}{14}}{\frac{1}{7}}=\frac{5\pi}{2}$

Then multiply by $\displaystyle \frac{180^{\circ}}{\pi}$:

$\displaystyle \frac{5\pi}{2}\times \frac{180^{\circ}}{\pi}=\frac{5\times 180^{\circ}}{2}=450^{\circ}$

Recall the definition of "radians" derived from the unit circle:

$\displaystyle 180$^{\circ}$ = \pi rad$

The quantity of radians given in the problem is $\displaystyle \frac{7\pi}{4}$. All that is required to convert this measure into degrees is to denote the unknown angle measure in degrees by $\displaystyle \Theta$ and set up a proportion equation using the aforementioned definition relating radians to degrees:

$\displaystyle \frac{180^{\circ}}{\Theta} = \frac{\pi rad}{\frac{7\pi}{4} rad}$

Cross-multiply the denominators in these fractions to obtain:

$\displaystyle 1260^{\circ}\pi rad=4\Theta\pi rad$

or

$\displaystyle 315^{\circ}\pi rad =\Theta\pi rad$.

Canceling like terms in these equations yields

$\displaystyle \Theta = 315^{\circ}$

Hence, the correct angle measure of $\displaystyle \frac{7\pi}{4}$ in degrees is $\displaystyle 315^{\circ}$.

1 radian is equal to $\displaystyle \frac{180}{\pi}$ degrees. Using this conversion factor,

$\displaystyle \frac{37\pi}{18}\times\frac{180}{\pi}=37\times10=370$.

We know that:

$\displaystyle 360^{\circ}=2\pi$ Radians

since the giving angle was in degrees then we multiply

$\displaystyle 45^{\circ}*\frac{\pi}{180^{\circ}}= \frac{\pi}{4}$

To convert degrees to radians, we need to multiply the given degree by $\displaystyle \frac{\pi}{180^{\circ}}$.

$\displaystyle 165*^{\circ}\frac{\pi}{180^{\circ}}=\frac{165\pi}{180}$

To simplify, we get:

$\displaystyle \frac{11\pi}{12}$