Question

figure lmno is a parallelogram. what is the value of x? 8 10 13 30

Explanation:

Step1: Recall parallelogram property

In a parallelogram, alternate interior angles (or consecutive angles? Wait, actually, in a parallelogram, opposite sides are parallel, so consecutive angles? Wait, no, looking at the angles: angle at M is \(2x\) and angle at N is \(6x - 40\). Wait, in a parallelogram, adjacent angles? Wait, no, actually, in a parallelogram, when two sides are parallel (like LM and ON, and MN and LO), so the angles at M and N: wait, maybe they are alternate interior angles? Wait, no, let's think again. In a parallelogram, the sides LM and ON are parallel, and MN is a transversal. So the angles at M and N: angle \(2x\) and angle \(6x - 40\) should be equal? Wait, no, maybe they are alternate interior angles? Wait, no, if LM is parallel to ON, and MN is the transversal, then angle at M and angle at N: wait, maybe they are equal? Wait, no, in a parallelogram, consecutive angles are supplementary, but if these are alternate interior angles, they should be equal. Wait, maybe the figure is such that LM is parallel to ON, and MN is the side, so angle at M ( \(2x\)) and angle at N ( \(6x - 40\)) are alternate interior angles, so they are equal. So set \(2x = 6x - 40\)? Wait, no, that would give negative x. Wait, maybe I got the angles wrong. Wait, maybe it's a parallelogram, so opposite angles are equal, or consecutive angles are supplementary. Wait, maybe the angles are equal. Wait, let's check the options. If x=10, then 2x=20, 6x-40=60-40=20. Oh! So 2x = 6x - 40? Wait, no, 2x=20, 6x-40=20 when x=10. Wait, 6x - 40 = 2x? Then 6x - 2x = 40 → 4x=40 → x=10. Yes, that works. So step 1: In a parallelogram, alternate interior angles (or equal angles) are equal, so set \(2x = 6x - 40\)? Wait, no, 6x - 40 = 2x? Wait, 6x - 2x = 40 → 4x=40 → x=10.

Step2: Solve for x

Set the two angles equal (since they are alternate interior angles in the parallelogram, so \(2x = 6x - 40\)? Wait, no, 6x - 40 = 2x? Wait, 6x - 2x = 40 → 4x = 40 → x = 10.

Answer:

10 (corresponding to the option "10")