Question

for what value of y must lmnp be a parallelogram?
lmnp must be a parallelogram for y = \square.

Explanation:

Step1: Recall parallelogram angle property

In a parallelogram, consecutive angles are supplementary (sum to \(180^\circ\)), and opposite angles are equal. Also, consecutive angles along a side should satisfy the supplementary property. But here, looking at the angles, we can use the property that consecutive angles between parallel sides are supplementary. Wait, actually, in a parallelogram, adjacent angles (consecutive angles) are supplementary. But also, opposite angles are equal. Wait, in the figure, angle at L is \(68^\circ\), angle at N is \(68^\circ\), so they are opposite angles (since in parallelogram LMNP, L and N are opposite vertices, M and P are opposite? Wait, maybe the sides are LM, MN, NP, PL. So angle at L and angle at N: if LM is parallel to NP and LN is a transversal? Wait, maybe better to use the property that consecutive angles are supplementary. Wait, no, let's think again. In a parallelogram, consecutive angles (e.g., angle at L and angle at M, angle at M and angle at N, etc.) are supplementary. But also, opposite angles are equal. Wait, in the given figure, angle at L is \(68^\circ\), angle at N is \(68^\circ\), so they are opposite angles (so that's consistent with opposite angles equal). Then angle at M and angle at P (which is \(y^\circ\)) should be equal? Wait, no, maybe angle at L and angle at P are consecutive? Wait, maybe the sides are L to M, M to N, N to P, P to L. So angle at L (between L-P and L-M) is \(68^\circ\), angle at N (between N-M and N-P) is \(68^\circ\), angle at M (between M-L and M-N) and angle at P (between P-N and P-L) should be equal. But also, consecutive angles: angle at L and angle at M should be supplementary, angle at M and angle at N should be supplementary, etc. Wait, maybe I made a mistake. Wait, the problem is to find y such that LMNP is a parallelogram. So in a parallelogram, consecutive angles are supplementary. Wait, angle at L is \(68^\circ\), angle at P (which is \(y^\circ\)): if LP is parallel to MN, then angle at L and angle at M are supplementary, angle at M and angle at N are supplementary, etc. Wait, maybe the correct approach is: in a parallelogram, consecutive angles are supplementary. Wait, no, let's look at the angles. The angle at L is \(68^\circ\), angle at N is \(68^\circ\), so they are opposite angles (so that's good, opposite angles equal). Then angle at M and angle at P (y) should be equal, and angle at L and angle at M should be supplementary. Wait, angle at L is \(68^\circ\), so angle at M should be \(180 - 68 = 112^\circ\)? No, that can't be. Wait, maybe I got the angles wrong. Wait, the figure: L, M, N, P. So sides: LM, MN, NP, PL. So angle at L: between PL and LM, angle at N: between MN and NP. So if LM is parallel to NP, then angle at L and angle at P are consecutive angles (since PL is a transversal), so they should be supplementary? Wait, no, PL and MN are parallel? Wait, maybe the correct property is that in a parallelogram, consecutive angles (adjacent angles) are supplementary. So angle at L (68°) and angle at P (y°) are consecutive? No, angle at L and angle at M are consecutive. Wait, maybe the problem is that angle at L and angle at P are same-side interior angles? Wait, maybe I need to re-express. Let's recall: In a parallelogram, opposite angles are equal, and consecutive angles are supplementary. So if angle at L is 68°, then angle at N (opposite) is 68°, and angle at M and angle at P (the other two angles) are equal, and each is supplementary to 68°. Wait, no, consecutive angles: angle at L and angle at M are consecutive, so they should add to 180°. So angle at M = 180 - 68 = 112°? But that contradicts. Wait, maybe the figure is labeled as L, M, N, P in order, so LM is adjacent to MN, MN to NP, NP to PL, PL to LM. So angle at L: between PL and LM (68°), angle at M: between LM and MN, angle at N: between MN and NP (68°), angle at P: between NP and PL (y°). Now, since LM is parallel to NP (opposite sides of parallelogram), and PL is a transversal, then angle at L (68°) and angle at P (y°) are same-side interior angles? Wait, no, same-side interior angles are supplementary. Wait, angle at L and angle at P: if PL is parallel to MN, then angle at L and angle at M are same-side interior angles. Wait, I think I'm overcomplicating. Let's use the property that in a parallelogram, consecutive angles are supplementary. Wait, angle at L (68°) and angle at P (y°): are they consecutive? No, angle at L and angle at M are consecutive. Wait, maybe the correct approach is that opposite angles are equal, and consecutive angles are supplementary. So angle at L (68°) and angle at N (68°) are opposite (so that's good). Then angle at M and angle at P (y°) are opposite, so they should be equal. Also, angle at L (68°) and angle at M are consecutive, so they should be supplementary. So angle at M = 180 - 68 = 112°, so angle at P (y°) is also 112°? Wait, that makes sense. Wait, no, wait: in a parallelogram, consecutive angles (adjacent angles) are supplementary. So angle at L (between PL and LM) and angle at M (between LM and MN) are consecutive, so they add to 180°. So angle at M = 180 - 68 = 112°. Then angle at M and angle at P are opposite angles, so they should be equal. So angle at P (y°) = angle at M = 112°? Wait, but let's check again. Alternatively, since angle at L and angle at N are both 68°, which are opposite angles (so that's correct for a parallelogram), then angle at M and angle at P (y°) must be equal and supplementary to 68°. Wait, no, supplementary to 68° would be 112°, so y = 180 - 68? Wait, no, 180 - 68 is 112? Wait, 68 + 112 = 180. Yes. So angle at L (68°) and angle at P (y°): if they are consecutive angles, then they should be supplementary. Wait, maybe angle at L and angle at P are consecutive. Wait, maybe the sides are L-M, M-N, N-P, P-L. So angle at L is between P-L and L-M (68°), angle at P is between N-P and P-L (y°). So L-M is parallel to N-P (opposite sides), and P-L is a transversal. Then angle at L (68°) and angle at P (y°) are same-side interior angles, which should be supplementary. Wait, same-side interior angles are supplementary when lines are parallel. So since L-M || N-P and P-L is transversal, angle at L + angle at P = 180°. Wait, that would mean 68 + y = 180, so y = 180 - 68 = 112. Wait, but earlier I thought angle at M and angle at P are equal. Wait, maybe angle at M is equal to angle at P, and angle at L is equal to angle at N. So angle at L = angle at N = 68°, angle at M = angle at P = y. Then since consecutive angles are supplementary, angle at L + angle at M = 180°, so 68 + y = 180, so y = 112. Yes, that makes sense. So the key property is that in a parallelogram, consecutive angles are supplementary (sum to 180°). So angle at L (68°) and angle at P (y°) are consecutive angles (since they are adjacent to side P-L), so they must sum to 180°. Hence, y = 180 - 68 = 112.

Step2: Calculate y

Using the property that consecutive angles in a parallelogram are supplementary:
\( y + 68 = 180 \)
Subtract 68 from both sides:
\( y = 180 - 68 \)
\( y = 112 \)

Answer:

\( 112 \)