Question

analyze the diagram below and complete the instructions that follow.
if ( mangle k = 82^{circ} ), find ( mangle l, mangle m ), and ( mangle n ).
a. ( mangle l = 82^{circ}, mangle m = 82^{circ}, mangle n = 98^{circ} )
b. ( mangle l = 82^{circ}, mangle m = 98^{circ}, mangle n = 98^{circ} )
c. ( mangle l = 98^{circ}, mangle m = 82^{circ}, mangle n = 98^{circ} )
d. ( mangle l = 98^{circ}, mangle m = 98^{circ}, mangle n = 82^{circ} )

Explanation:

Step1: Properties of a parallelogram

In a parallelogram \(KLMN\), \(KL\parallel MN\) and \(KM\parallel LN\). Opposite angles of a parallelogram are equal. So \(m\angle K=m\angle M\) and \(m\angle L = m\angle N\). Also, consecutive angles of a parallelogram are supplementary. That is \(m\angle K + m\angle L=180^{\circ}\).

Step2: Find \(m\angle L\)

Given \(m\angle K = 82^{\circ}\). Using the supplementary - angle property \(m\angle K+m\angle L = 180^{\circ}\). Substitute \(m\angle K = 82^{\circ}\) into the equation: \(82^{\circ}+m\angle L=180^{\circ}\). Then \(m\angle L=180^{\circ}- 82^{\circ}=98^{\circ}\).

Step3: Find \(m\angle M\) and \(m\angle N\)

Since \(m\angle K=m\angle M\) (opposite angles of a parallelogram), \(m\angle M = 82^{\circ}\). And since \(m\angle L=m\angle N\) (opposite angles of a parallelogram), \(m\angle N=98^{\circ}\).

Answer:

C. \(m\angle L = 98^{\circ},m\angle M = 82^{\circ},m\angle N = 98^{\circ}\)