In a rhombus, the diagonals bisect each other at right angles. Since you have rhombus \(LMNO\) with diagonal \(LN\), and angle \(MNL\) is \(55^\circ\), we can find angle \(N\) using the properties of the rhombus.
First, note that:
- \(MNL\) is one of the angles formed by the intersection of diagonal \(LN\) with side \(LM\).
Since diagonals bisect the angles of the rhombus:
- Triangle \(MNL\) is formed, and angles \(MNL\) and \(NLM\) are adjacent angles:
- Therefore, \(NLM = 55^\circ\) as well.
Now, in triangle \(MNL\):
- The angle sum property of a triangle states that the sum of the angles is \(180^\circ\): \[ \angle MNL + \angle NLM + \angle LMN = 180^\circ \] Substituting the known angles: \[ 55^\circ + 55^\circ + \angle LMN = 180^\circ \] \[ 110^\circ + \angle LMN = 180^\circ \] \[ \angle LMN = 180^\circ - 110^\circ = 70^\circ \]
Now, since in rhombus \(LMNO\), opposite angles are equal:
- Angle \(N\) (which is the same as angle \(LMN\)) is \(70^\circ\).
Thus, the measure of angle \(N\) is \(70^\circ\).