Question

applying properties to find missing measures
for rhombus lmno, ( mangle lon = 102^{circ} ) and ( np = 5 ) units.
use the diagram of rhombus lmno to find the missing measures.
the measure of ( angle lpm ) is ( 90 )
the measure of ( angle pmn ) is
the length of ( ln ) is ( 10 ) units.
feedback
the diagonals of a rhombus are angle bisectors for the angles at the vertices, and they are perpendicular bisectors of each other.

Explanation:

Step1: Recall properties of rhombus

In a rhombus, the diagonals are perpendicular bisectors of each other. So, \(\angle LPM = 90^{\circ}\). Also, the diagonals bisect the angles of the rhombus.

Step2: Find \(\angle PMN\)

Given \(m\angle LON=102^{\circ}\). Since the diagonals of a rhombus bisect the vertex angles, \(\angle ONM=\angle OMN\). In \(\triangle ONM\), \(m\angle LON\) is an exterior angle. But we can also use the fact that the diagonals bisect the angles. The adjacent angles of a rhombus are supplementary. Let's use the angle - bisecting property. \(\angle LON\) and \(\angle LMN\) are related. Since the diagonal \(LN\) bisects \(\angle LMN\) and \(\angle LON\) and \(\angle LMN\) are related as \(m\angle LON + m\angle LMN= 180^{\circ}\) (adjacent angles of a rhombus are supplementary). So \(m\angle LMN=78^{\circ}\). Since the diagonal \(MN\) bisects \(\angle LMN\), \(m\angle PMN=\frac{1}{2}m\angle LMN\). So \(m\angle PMN = 39^{\circ}\)

Step3: Find length of \(LN\)

Given \(NP = 5\) units. Since the diagonals of a rhombus bisect each other, \(LN=2\times NP\). So \(LN = 10\) units

Answer:

The measure of \(\angle PMN\) is \(39^{\circ}\)