Rhombus LMNO

Question

has a diagonal line LN¯¯¯¯¯¯¯¯
, and ∠MNL
is 55 degrees. What is m∠MNO
?(1 point)
m∠N=
°

Answer 1

In a rhombus, the diagonals bisect each other at right angles and also bisect the angles at each vertex.

Given that $\mathrm{\angle }MNL={55}^{\circ }$, this angle is split into two equal angles by diagonal $LN$. Therefore, we can find $\mathrm{\angle }NML$:

$$\mathrm{\angle }NML=\mathrm{\angle }MNL={55}^{\circ }$$

Now, since $\mathrm{\angle }MNL$ and $\mathrm{\angle }NML$ are angles associated with vertex $N$, the remaining angle $\mathrm{\angle }N$ (the angles at vertex $N$ split by the diagonal) will be:

$$\mathrm{\angle }N=\mathrm{\angle }MNL+\mathrm{\angle }NML={55}^{\circ }+{55}^{\circ }={110}^{\circ }$$

Since the diagonals of a rhombus bisect the angles, both angles $MNO$ and $NMO$ must be equal to $\mathrm{\angle }N$ but on the opposite side, and so:

$$\mathrm{\angle }MNO=\mathrm{\angle }MNL={55}^{\circ }$$

Thus,

$$m\mathrm{\angle }MNO={55}^{\circ }$$