Angles LMN and OMP have the following measures:
m∠LMN = (x + 12)°, m∠OMP = (4x − 7)°
Part A: If angle LMN and angle OMP are complementary angles, find the value of x. Show every step of your work. (4 points)
Part B: Use the value of x from Part A to find the measures of angles LMN and OMP. Show every step of your work. (4 points)
Part C: Could the angles also be vertical angles? Explain. (4 points)
Part A:
If angle LMN and angle OMP are complementary angles, then their measures sum up to 90°. Thus, we have the equation:
m∠LMN + m∠OMP = 90°
Substituting the given measures, we have:
(x + 12)° + (4x - 7)° = 90°
Combining like terms:
5x + 5° = 90°
Subtracting 5° from both sides of the equation:
5x = 85°
Dividing both sides by 5:
x = 17°
Therefore, the value of x is 17°.
Part B:
Using the value of x from Part A, we can find the measures of angles LMN and OMP.
m∠LMN = (x + 12)° = 17° + 12° = 29°
m∠OMP = (4x - 7)° = 4(17°) - 7° = 68° - 7° = 61°
Therefore, the measure of angle LMN is 29° and the measure of angle OMP is 61°.
Part C:
No, the angles LMN and OMP cannot be vertical angles. Vertical angles are formed when two lines intersect, and their measures are always equal. In this case, there is no indication of intersecting lines, so the angles cannot be vertical angles.