Question

Rhombus LMNO has a diagonal line MO¯¯¯¯¯¯¯¯¯, with m∠M=74°. What is m∠LOM?(1 point)
Responses which answer is correct and why

106°
106°

53°
53°

74°
74°

37°

Answer 1

In a rhombus, the diagonals bisect each other and the angles. Since \( MO \) is a diagonal of rhombus \( LMNO \), it divides the rhombus into two triangles: \( LMO \) and \( NMO \).

Given that \( \angle M = 74^\circ \), we can find \( \angle LMO \). In triangle \( LMO \):

\[
\angle L + \angle M + \angle LMO = 180^\circ
\]

Since \( \angle LMO \) and \( \angle L \) are opposite angles created by the diagonals, they are equal. Therefore:

\[
2\angle LMO + 74^\circ = 180^\circ
\]

This can be simplified to:

\[
2\angle LMO = 180^\circ - 74^\circ
\]
\[
2\angle LMO = 106^\circ
\]
\[
\angle LMO = 53^\circ
\]

Now, since angles \( LMO \) and \( L \) are equal as the diagonals bisect the angles of the rhombus, we have:

\[
\angle L = 53^\circ
\]

Next, we look for \( \angle LOM \) which is in triangle \( LMO \):

\[
\angle LOM + \angle M + \angle L = 180^\circ
\]
\[
\angle LOM + 74^\circ + 53^\circ = 180^\circ
\]
\[
\angle LOM + 127^\circ = 180^\circ
\]
\[
\angle LOM = 180^\circ - 127^\circ = 53^\circ
\]

Thus, \( m\angle LOM \) is \( 106^\circ \) because it is supplementary to angle \( M \) and \( L \).

Therefore, the correct answer is:
**106°**