Question

in the rectangle below, $jn = 4x + 5$, $jl = 5x + 28$, and $m\angle knl = 54^\circ$. find the value of $x$ and $m\angle nmj$.

Explanation:

Step1: Recall rectangle diagonals property

In a rectangle, diagonals are equal and bisect each other. So \( JL = 2JN \).
Given \( JN = 4x + 5 \) and \( JL = 5x + 28 \), substitute into the equation:
\( 5x + 28 = 2(4x + 5) \)

Step2: Solve for \( x \)

Expand the right - hand side: \( 5x + 28 = 8x + 10 \)
Subtract \( 5x \) from both sides: \( 28 = 3x + 10 \)
Subtract 10 from both sides: \( 18 = 3x \)
Divide both sides by 3: \( x = 6 \)

Step3: Analyze \( \angle NMJ \)

In rectangle \( JMLK \), \( \angle KNL = 54^{\circ} \). Since \( \angle KNL \) and \( \angle JNM \) are vertical angles, \( \angle JNM = 54^{\circ} \).
Also, in triangle \( NMJ \), \( JN = MN \) (diagonals bisect each other), so triangle \( NMJ \) is isosceles with \( \angle NMJ=\angle NJM \).
The sum of angles in a triangle is \( 180^{\circ} \), so \( \angle NMJ=\frac{180^{\circ}-\angle JNM}{2} \)
Substitute \( \angle JNM = 54^{\circ} \): \( \angle NMJ=\frac{180 - 54}{2}=\frac{126}{2}=63^{\circ} \)

Answer:

\( x = 6 \)
\( m\angle NMJ = 63^{\circ} \)