Answer
$(a)$ "The measure of an angle is the amount of rotation about the vertex required to move $R_1$ onto $R_2$ (address to the image below). This is how much the angle opens. One unit of measurement for angles is the degree. An angle of measure $1$ degree is formed by rotating the initial side $\frac{1}{360}$ of a complete revolution."
Simply saying, angle measure is a unit of measurement of a rotation where full rotation is $360°$.
$(b)$ "If a circle of radius $1$ is drawn with the vertex of an angle at its center, then the measure of this angle in radians is the length of the arc that subtends the angle"
$(c)$ We have to multiply by $\frac{\pi}{180}$
$\frac{\pi}{4}$
$(d)$ We have to multiply by $\frac{180}{\pi}$
$\frac{360}{\pi}\approx 114.6°$
Work Step by Step
$(a)$ As explained in the chapter $6.1$ "the measure of an angle is the amount of rotation about the vertex required to move $R_1$ onto $R_2$ (address to the image below). This is how much the angle opens. One unit of measurement for angles is the degree. An angle of measure $1$ degree is formed by rotating the initial side $\frac{1}{360}$ of a complete revolution."
Simply saying, angle measure is a unit of measurement of a rotation.
$(b)$ Radian measurement is often used in mathematics. As explained in the chapter previously, "the amount an angle opens is measured along the arc of a circle of radius $1$ with its center at vertex of the angle."
For a circle with radius $1$, the whole circumference is $2\pi$. So $360$ degrees correspond to $2\pi$ (So, $180°$ is $\pi$ and $90°$ is $\frac{\pi}{2}$).
$(c)$ To convert degrees to radians, we can use the relation mentioned in $(b)$. That is multiply by $\frac{\pi}{180}$
$45\times\frac{\pi}{180}=\frac{\pi}{4}$
$(d)$ To convert radians to degrees, we can use the relation mentioned above. That is multiply by $\frac{180}{\pi}$
$2\times \frac{180}{\pi}=\frac{360}{\pi}\approx 114.6°$
