Question

1. given the rhombus ( j k l m, m k = 30 ), ( n l = 13 ), and ( m angle m k l = 41^{circ} ). determine the missing measures.

( n k = )( m angle j m l = )( j l = )( m angle m l k = )( k l = )( m angle m n l = )( m angle j k m = )( m angle k j l = )

Explanation:

Step1: Use the property of rhombus diagonals bisect each other

In a rhombus, the diagonals bisect each other. Given \(MK = 30\), then \(NK=\frac{MK}{2}\).
\(NK=\frac{30}{2}=15\)

Step2: Calculate \(JL\)

Since \(NL = 13\) and \(JL = 2\times NL\) (diagonals bisect each other), then \(JL=2\times13 = 26\)

Step3: Use Pythagorean theorem to find \(KL\)

In right - triangle \(NKL\) (diagonals of a rhombus are perpendicular), by the Pythagorean theorem \(KL=\sqrt{NK^{2}+NL^{2}}\). Substitute \(NK = 15\) and \(NL = 13\)
\(KL=\sqrt{15^{2}+13^{2}}=\sqrt{225 + 169}=\sqrt{394}\approx19.85\)

Step4: Find \(m\angle JKM\)

Given \(m\angle MKL=41^{\circ}\), and in a rhombus \(m\angle JKM = m\angle MKL\) (diagonals bisect the angles of the rhombus), so \(m\angle JKM = 41^{\circ}\)

Step5: Find \(m\angle JML\)

In right - triangle \(JNM\), \(m\angle MJN=90^{\circ}-m\angle JKM\). Since \(m\angle JKM = 41^{\circ}\), then \(m\angle MJN = 49^{\circ}\). In \(\triangle JML\), \(m\angle JML=2\times m\angle MJN\) (diagonals bisect the angles of the rhombus)
\(m\angle JML = 98^{\circ}\)

Step6: Find \(m\angle MLK\)

In a rhombus, adjacent angles are supplementary. Let \(m\angle MLK=x\), and \(m\angle JML\) and \(m\angle MLK\) are adjacent. So \(x + 98^{\circ}=180^{\circ}\), then \(x=82^{\circ}\), \(m\angle MLK = 82^{\circ}\)

Step7: Find \(m\angle MNL\)

Since the diagonals of a rhombus are perpendicular, \(m\angle MNL = 90^{\circ}\)

Step8: Find \(m\angle KJL\)

\(m\angle KJL=m\angle MJN\) (diagonals bisect the angles of the rhombus), so \(m\angle KJL = 49^{\circ}\)

Answer:

\(NK = 15\), \(JL = 26\), \(KL=\sqrt{394}\approx19.85\), \(m\angle JKM = 41^{\circ}\), \(m\angle JML = 98^{\circ}\), \(m\angle MLK = 82^{\circ}\), \(m\angle MNL = 90^{\circ}\), \(m\angle KJL = 49^{\circ}\)