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.
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.
Part C: Could the angles also be vertical angles? Explain.
Part A:
If angle LMN and angle OMP are complementary angles, their sum should be 90 degrees.
So, we have the equation:
(x + 12)° + (4x - 7)° = 90°
Combining like terms, we get:
5x + 5° = 90°
Subtracting 5° from both sides, we have:
5x = 85°
Dividing both sides by 5, we get:
x = 17°
Part B:
Now that we have found the value of x to be 17°, we can substitute it back into the equations to find the measures of angles LMN and OMP.
m∠LMN = (x + 12)°
m∠LMN = (17 + 12)°
m∠LMN = 29°
m∠OMP = (4x - 7)°
m∠OMP = (4 * 17 - 7)°
m∠OMP = 61°
So, the measure of angle LMN is 29° and the measure of angle OMP is 61°.
Part C:
No, the angles could not be vertical angles. Vertical angles are formed when two lines intersect and the angles opposite each other are congruent. In this case, angles LMN and OMP are complementary, not congruent, so they cannot be vertical angles.