Question

find the indicated angle measures in the given rhombus.
$m\angle mpn$
$m\angle pmn$
$m\angle mnp$
$m\angle mno$

Explanation:

Step1: Find $\angle MPN$

In a rhombus, diagonals are perpendicular, so they form right angles.
$\angle MPN = 90^\circ$

Step2: Find $\angle PMN$

First, identify $\angle POM = 62^\circ$. In a rhombus, diagonals bisect vertex angles, so $\angle MLO = \angle OMN = 62^\circ$. In right triangle $MPN$, the acute angles sum to $90^\circ$.
$\angle PMN = 90^\circ - 62^\circ = 28^\circ$

Step3: Find $\angle MNO$

In a rhombus, opposite angles are equal, and consecutive angles are supplementary. First, find $\angle MLO = 62^\circ$, so the full vertex angle $\angle MLO = 2\times62^\circ = 124^\circ$. $\angle MNO$ is consecutive to $\angle MLO$, so:
$\angle MNO = 180^\circ - 124^\circ = 56^\circ$

Step4: Find $\angle MLO$

Diagonals bisect vertex angles, so $\angle MLO = 2\times\angle PLO = 2\times62^\circ = 124^\circ$

Answer:

$m\angle MPN = 90^\circ$
$m\angle PMN = 28^\circ$
$m\angle MNO = 56^\circ$
$m\angle MLO = 124^\circ$