Question

in the rectangle below, $jn = 3x - 6$, $km = 36$, and $mangle nlm = 42^{circ}$. find the value of $x$ and $mangle nkl$.

Explanation:

Step1: Use rectangle property of diagonals

In a rectangle, the diagonals are equal and bisect each other. So $JN=\frac{1}{2}KM$. Given $KM = 36$, then $JN=\frac{36}{2}=18$.

Step2: Solve for x

Since $JN = 3x - 6$ and $JN=18$, we set up the equation $3x - 6=18$. Add 6 to both sides: $3x=18 + 6=24$. Divide both sides by 3: $x=\frac{24}{3}=8$.

Step3: Find angle measure

In a rectangle, each angle is a right - angle, so $\angle KML = 90^{\circ}$. In $\triangle NLM$, we know $\angle NLM=42^{\circ}$. Then $\angle NML=90^{\circ}-\angle NLM = 90^{\circ}-42^{\circ}=48^{\circ}$. Since the diagonals of a rectangle are equal and bisect each other, $NK = NL$, so $\triangle NKL$ is isosceles. And $\angle NKL=\angle NLK$. In right - triangle $KML$, $\angle NKL=\angle NLK = 48^{\circ}$.

Answer:

$x = 8$
$m\angle NKL=48^{\circ}$