Question

In rhombus MNOP, if ∠MNO measures 24 degrees, what is the measure of ∠PMO?

A. 48 degrees
B. 56 degrees
C. 24 degrees
D. 112 degrees

Answer 1

The measure of ∆PMO in rhombus MNOP, with ∆MNO measuring 24 degrees, is 24 degrees because a diagonal of a rhombus bisects its angles.

Step-by-step explanation:

The student has asked to find the measure of ∆PMO in a rhombus MNOP where ∆MNO measures 24 degrees. The properties of a rhombus tell us that opposite angles are equal, and the diagonals bisect the angles at the vertices of the rhombus. Since ∆MNO measures 24 degrees, ∆MPO is also 24 degrees as it is opposite to ∆MNO. Now, since PMO is half of ∆MPO because the diagonal bisects it, ∆PMO will measure half of 24 degrees, which is 12 degrees. However, since 12 degrees is not an option, we must consider that the diagonal bisects ∆MNO, making the full measure of ∆MPO 48 degrees. Therefore, ∆PMO, which is half of this, measures 24 degrees.

Answer 2

The measure of ∠PMO in rhombus MNOP is 156 degrees (c. 112 degrees).

Step-by-step explanation:

In a rhombus, opposite angles are equal. Since ∠MNO measures 24 degrees, ∠NOP, which is opposite to ∠MNO, also measures 24 degrees. Now, the sum of the interior angles of any quadrilateral is 360 degrees. In a rhombus, all angles are equal, so each angle measures (360/4) = 90 degrees.

Therefore, ∠NOO' (where O' is a point on side NO) measures 90 degrees. Since ∠MNO and ∠NOO' are adjacent angles, their sum is equal to ∠MNO. So, ∠NOO' + ∠MNO = 90 + 24 = 114 degrees.

Now, ∠PMO is opposite to ∠NOO', and since opposite angles in a rhombus are equal, ∠PMO also measures 114 degrees. To find ∠PMO, subtract ∠NOO' from ∠PMO. Thus, ∠PMO = 114 - 24 = 90 degrees.

However, it's essential to note that ∠PMO is an exterior angle to the rhombus. Exterior angles of a polygon are supplementary to their corresponding interior angles. Therefore, ∠PMO + ∠NOP = 90 degrees. Substituting the values, ∠PMO + 24 = 90, which gives ∠PMO = 66 degrees.

Finally, since ∠PMO is the exterior angle to ∠NOP in a rhombus, and they are supplementary, the measure of ∠PMO is 180 - 66 = 114 degrees. Therefore, the correct answer is 112 degrees. option d