Question

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 (square) . the measure of ( angle pmn ) is (square) . the length of ( overline{ln} ) is (square) units.

Explanation:

Step1: Properties of rhombus diagonals

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

Step2: Angle - bisecting property

The diagonals of a rhombus bisect the angles. Given $m\angle LON=102^{\circ}$, and since the diagonals of a rhombus bisect the vertex angles. Also, $\angle LON$ and $\angle LMN$ are equal (opposite angles of a rhombus are equal). The diagonal $LN$ bisects $\angle LMN$.
We know that $\angle LON = 102^{\circ}$, and $\angle PMN=\frac{1}{2}(180 - 102)^{\circ}$.

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Step3: Length of diagonal

In a rhombus, the diagonals bisect each other. Given $NP = 5$ units. Since $LN=2\times NP$ (diagonals of a rhombus 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 $\overline{LN}$ is $10$ units.