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
the measure of ( angle lpm ) is
the measure of ( angle pmn ) is
the length of ( ln ) is units.

Explanation:

Step1: Recall properties of a rhombus

In a rhombus, the diagonals are perpendicular to each other. So, $\angle LPM = 90^{\circ}$.

Step2: Use angle - sum property and properties of rhombus diagonals

The diagonals of a rhombus bisect the angles. Given $m\angle LON=102^{\circ}$, then $m\angle PON=\frac{1}{2}m\angle LON = 51^{\circ}$. In right - triangle $PMN$ (since diagonals are perpendicular), $\angle PMN = 90^{\circ}-\angle PON$. So, $m\angle PMN=90 - 51=39^{\circ}$.

Step3: Use the property of rhombus diagonals bisecting each other

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

Answer:

The measure of $\angle LPM$ is $90^{\circ}$. The measure of $\angle PMN$ is $39^{\circ}$. The length of $LN$ is $10$ units.