Question

Question
Rhombus LMNO has a diagonal line MO¯¯¯¯¯¯¯¯¯, with m∠M=74°. What is m∠LOM?(1 point)
Responses a b c or d which one is it and why

106°
106°

53°
53°

74°
74°

37°
37°
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Answer 1

In a rhombus, the diagonals bisect each other at right angles, meaning they create four right triangles.

Given that \( m\angle M = 74^\circ \), we can analyze triangle \( LOM \):

1. Since \( MO \) is a diagonal, \( \angle LOM \) is opposite \( \angle M \).
2. In triangles formed by the diagonals, opposite angles are equal, hence \( \angle LOM \) (which is the angle opposite to \( \angle M \)) also helps in finding the value of the angles.
3. We know that the angles in a triangle sum to \( 180^\circ \). Therefore, if we denote the angles at \( O \) (which are both \( \angle LOM \) and \( \angle NOL \)) and sum it with \( \angle M \):

\[
m\angle LOM + m\angle M + m\angle NOL = 180^\circ
\]

Since \( \angle LOM \) and \( \angle NOL \) are equal (as diagonals bisect angles):

\[
2 \cdot m\angle LOM + 74^\circ = 180^\circ
\]

Now we can solve for \( m\angle LOM \):

\[
2 \cdot m\angle LOM = 180^\circ - 74^\circ
\]
\[
2 \cdot m\angle LOM = 106^\circ
\]
\[
m\angle LOM = 53^\circ
\]

So the correct response is **53°**.

Thus, the answer is **(b) 53°**.