Question

for rhombus lmno, m∠lon = 102° and np = 5 units. use the diagram of rhombus lmno to find the missing measures. the measure of ∠lpm is the measure of ∠pmn is the length of ln is units. 5 10 25 102

Explanation:

Step1: Recall property of rhombus diagonals

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

Step2: Find half - angle of $\angle LON$

Since the diagonals of a rhombus bisect the angles of the rhombus, $\angle PON=\frac{1}{2}\angle LON$. Given $\angle LON = 102^{\circ}$, then $\angle PON = 51^{\circ}$. In right - triangle $PMN$, $\angle PMN$ and $\angle PON$ are alternate interior angles (because of the parallel sides of the rhombus). So, $\angle PMN= 51^{\circ}$.

Step3: Recall property of rhombus diagonals for length

The diagonals of a rhombus bisect each other. Given $NP = 5$ units, and $LN$ is a diagonal with $P$ as the mid - point. So, $LN=2\times NP$. Then $LN = 10$ units.

Answer:

The measure of $\angle LPM$ is $90^{\circ}$.
The measure of $\angle PMN$ is $51^{\circ}$.
The length of $LN$ is $10$ units.