Find an angle in each quadrant with a common reference angle with ${296}^{\circ }$, from $0^{\circ }\le \theta <{360}^{\circ }$

Step 1 ：Given the angle $\theta ={296}^{\circ }$, which is in the fourth quadrant.

Step 2 ：To find the reference angle, subtract the given angle from 360 degrees. So, the reference angle is ${360}^{\circ }-{296}^{\circ }={64}^{\circ }$.

Step 3 ：An angle in the first quadrant with the same reference angle is simply the reference angle itself, so ${\text{angle}}_{q1}={64}^{\circ }$.

Step 4 ：An angle in the second quadrant with the same reference angle is found by subtracting the reference angle from 180 degrees, so ${\text{angle}}_{q2}={180}^{\circ }-{64}^{\circ }={116}^{\circ }$.

Step 5 ：An angle in the third quadrant with the same reference angle is found by adding the reference angle to 180 degrees, so ${\text{angle}}_{q3}={180}^{\circ }+{64}^{\circ }={244}^{\circ }$.

Step 6 ：An angle in the fourth quadrant with the same reference angle is the given angle, so ${\text{angle}}_{q4}={296}^{\circ }$.

Step 7 ：Final Answer: The angles in each quadrant with a common reference angle with ${296}^{\circ }$ are ${64}^{\circ }$, ${116}^{\circ }$, ${244}^{\circ }$, and ${296}^{\circ }$. So, the final answer is $\boxed{{\displaystyle {64}^{\circ },{116}^{\circ },{244}^{\circ },{296}^{\circ }}}$.