Question

in parallelogram lmno, what is the measure of angle n? 50° 70° 110° 130°

Explanation:

Step1: Recall parallelogram angle property

In a parallelogram, consecutive angles are supplementary (sum to \(180^\circ\)). So, \(\angle L + \angle O = 180^\circ\).
Given \(\angle L=(2x + 10)^\circ\) and \(\angle O=(x + 20)^\circ\), we set up the equation:
\((2x + 10)+(x + 20)=180\)

Step2: Solve for \(x\)

Combine like terms: \(3x+30 = 180\)
Subtract 30 from both sides: \(3x=180 - 30=150\)
Divide by 3: \(x=\frac{150}{3}=50\)

Step3: Find \(\angle O\)

Substitute \(x = 50\) into \(\angle O=(x + 20)^\circ\): \(\angle O=(50 + 20)^\circ=70^\circ\)

Step4: Find \(\angle N\)

In a parallelogram, opposite angles are equal, and consecutive angles are supplementary. \(\angle N\) and \(\angle L\) are supplementary to \(\angle O\)? Wait, no: \(\angle N\) and \(\angle L\) are equal? Wait, no, in parallelogram \(LMNO\), \(\angle L\) and \(\angle N\) are... Wait, consecutive angles: \(\angle O\) and \(\angle N\) are consecutive? Wait, no, let's label the parallelogram: \(L - M - N - O - L\). So sides \(LM\parallel ON\) and \(LO\parallel MN\). So \(\angle L\) and \(\angle N\) are opposite? No, \(\angle L\) and \(\angle N\) are consecutive? Wait, no, in a parallelogram, consecutive angles (adjacent angles) are supplementary. So \(\angle O\) and \(\angle N\) are consecutive? Wait, \(O - N - M - L - O\), so \(\angle O\) and \(\angle N\) are adjacent? Wait, no, \(\angle O\) is at \(O\), between \(LO\) and \(ON\), \(\angle N\) is at \(N\), between \(ON\) and \(NM\). So \(LO\parallel NM\), so \(\angle O\) and \(\angle N\) are same - side interior angles? Wait, no, in parallelogram, \(\angle O\) and \(\angle N\) are supplementary? Wait, no, let's correct: In parallelogram \(LMNO\), \(\angle L\) and \(\angle O\) are consecutive (supplementary), \(\angle O\) and \(\angle N\) are consecutive (supplementary)? No, wait, opposite angles: \(\angle L=\angle N\)? No, wait, no: in a parallelogram, opposite angles are equal, and consecutive angles are supplementary. So \(\angle L=\angle N\)? Wait, no, \(\angle L\) and \(\angle M\) are supplementary, \(\angle M=\angle O\), \(\angle O\) and \(\angle N\) are supplementary, \(\angle N=\angle L\). Wait, let's re - express:

Since \(LO\parallel MN\), \(\angle O\) and \(\angle N\) are same - side interior angles? No, \(ON\) is a transversal. Wait, maybe easier: we found \(\angle O = 70^\circ\), and \(\angle N\) is supplementary to \(\angle O\)? Wait, no, \(\angle N\) and \(\angle L\) are equal? Wait, we found \(x = 50\), so \(\angle L=(2x + 10)^\circ=(2\times50+10)^\circ = 110^\circ\). Then, since \(\angle O = 70^\circ\), and \(\angle N\) is supplementary to \(\angle O\)? Wait, no, \(\angle N\) and \(\angle L\) are equal? Wait, no, in parallelogram, \(\angle L\) and \(\angle N\) are opposite? Wait, no, \(L\) and \(N\) are not opposite. Wait, the vertices are \(L\), \(M\), \(N\), \(O\) in order, so the angles are \(\angle L\) (at \(L\)), \(\angle M\) (at \(M\)), \(\angle N\) (at \(N\)), \(\angle O\) (at \(O\)). So \(\angle L\) and \(\angle M\) are consecutive, \(\angle M\) and \(\angle N\) are consecutive, \(\angle N\) and \(\angle O\) are consecutive, \(\angle O\) and \(\angle L\) are consecutive. So consecutive angles are supplementary. So \(\angle N\) and \(\angle O\) are consecutive? No, \(\angle N\) and \(\angle M\) are consecutive, \(\angle M\) and \(\angle L\) are consecutive. Wait, I think I made a mistake earlier. Let's use the correct property: in parallelogram \(LMNO\), \(\angle L\) and \(\angle N\) are equal? No, \(\angle L\) and \(\angle N\) are supplementary? Wait, no, let's calculate \(\angle L=(2x + 10)^\circ=(2\times50 + 10)^\circ=110^\circ\). Then, since \(\angle O = 70^\circ\), and \(\angle N\) is supplementary to \(\angle O\)? Wait, no, \(\angle N\) and \(\angle L\) are supplementary? No, \(\angle N\) and \(\angle O\) are supplementary? Wait, let's check the sides: \(LO\) is parallel to \(MN\), and \(ON\) is a transversal. So \(\angle O\) and \(\angle N\) are same - side interior angles, so they should be supplementary? Wait, no, same - side interior angles are supplementary when lines are parallel. So \(LO\parallel MN\), transversal \(ON\), so \(\angle O+\angle N = 180^\circ\)? Wait, no, \(LO\) and \(MN\) are parallel, \(ON\) is the transversal, so \(\angle O\) and \(\angle N\) are same - side interior angles, so \(\angle O+\angle N = 180^\circ\)? Wait, no, \(\angle O\) is at \(O\), between \(LO\) and \(ON\), \(\angle N\) is at \(N\), between \(ON\) and \(MN\). So \(LO\) and \(MN\) are parallel, so \(\angle O\) and \(\angle N\) are same - side interior angles, so they are supplementary. Wait, but \(\angle O = 70^\circ\), so \(\angle N=180^\circ - 70^\circ = 110^\circ\). Alternatively, since \(\angle L = 110^\circ\), and \(\angle N\) is equal to \(\angle L\)? Wait, no, in a parallelogram, opposite angles are equal. So \(\angle L\) and \(\angle N\) are opposite? Wait, \(L\) and \(N\) are opposite vertices? In parallelogram \(LMNO\), the opposite vertices are \(L\) and \(N\), \(M\) and \(O\). So \(\angle L=\angle N\), and \(\angle M=\angle O\). Oh! That's the key property: in a parallelogram, opposite angles are equal. So \(\angle L=\angle N\), \(\angle M=\angle O\). We found \(\angle L=(2x + 10)^\circ=(2\times50 + 10)^\circ = 110^\circ\), so \(\angle N=\angle L = 110^\circ\).

Answer:

\(110^\circ\) (corresponding to the option \(110^\circ\))