KM = 4x + 8 and m∠NKL = 56°, obtained by utilizing the properties of a rectangle and the tangent of the given angle.
Let's denote the lengths of the sides of the rectangle as follows: JN = 4x + 8, LN = 5x + 1, KM = KL = JM = JL (since JKLM is a rectangle).
Since JKLM is a rectangle, opposite sides are equal. Therefore, JM = KL and JN = KM. Given that m∠JNM = 124°, it indicates that JN and JM are adjacent sides, forming an angle at \(N\).
Now, since JN and JM are adjacent sides in a rectangle, they are also opposite sides of a right angle at N. Therefore, we can use the tangent of the angle m∠JNM to relate JN\) and JM:
tan(124°) = JN/JM
Solving for JM, we get JM = JN tan(124°).
Now, substitute the expressions for JN and JM:
JM = (4x + 8) tan(124°)
To find KM, we know that KM = JN, so KM = 4x + 8.
Additionally, m∠NKL is supplementary to m∠JNM since they form a straight line. Therefore, m∠NKL = 180° - 124° = 56°.