Question

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

106°
106°

53°
53°

74°
74°

37°

Answer 1

In a rhombus, the diagonals bisect each other at right angles. Therefore, since MO is a diagonal of rhombus LMNO, it bisects the angles at vertices L and N as well as M and O.

Given that \( m\angle M = 74^\circ \), we can find \( m\angle O \) since \( \angle M \) and \( \angle O \) are opposite angles in the rhombus and are equal:

\[
m\angle O = 74^\circ
\]

The diagonals also bisect the angles at vertices L and N. Since \( MO \) is the diagonal of LMNO, it splits \( \angle L \) and \( \angle N \) into two equal parts each.

Since the total of the angles in any quadrilateral is \( 360^\circ \):

\[
m\angle L + m\angle M + m\angle N + m\angle O = 360^\circ
\]

We know that \( m\angle L = m\angle N \) (since they are opposite angles in a rhombus), and \( m\angle M = m\angle O = 74^\circ \).

Let \( x = m\angle L = m\angle N \). Therefore, we can state:

\[
x + 74 + x + 74 = 360
\]
\[
2x + 148 = 360
\]
\[
2x = 360 - 148
\]
\[
2x = 212
\]
\[
x = 106
\]

Thus, \( m\angle L = 106^\circ \).

To find \( m\angle LOM \) (which is half of \( m\angle L \)), we divide by 2:

\[
m\angle LOM = \frac{m\angle L}{2} = \frac{106^\circ}{2} = 53^\circ
\]

So, the measure of \( m\angle LOM \) is \( 53^\circ \).

Therefore, the correct response is:

**53°**