Question

M and N are vertical angles where m ∠M = 4x + 14 and m ∠N = 6x – 24. Determine whether angles M and N are acute, obtuse, or right angles?

Answer 1

Both angles M and N are congruent right angles, as solving the given expressions for their measures yields 90 degrees for each.

Step-by-step explanation:

When M and N are vertical angles, they are congruent, which means they have equal measures. Therefore, we can set the expressions for m ∠M and m ∠N equal to each other and solve for x:

m ∠M = 4x + 14
m ∠N = 6x - 24
4x + 14 = 6x - 24

Let's solve for x:

4x - 6x = -24 - 14
-2x = -38
x = 19

Now, plug x back into the expressions:

m ∠M = 4(19) + 14 = 76 + 14 = 90 degrees
m ∠N = 6(19) - 24 = 114 - 24 = 90 degrees

Angles exactly 90 degrees are right angles. Hence, angles M and N are right angles.