Question

figure abcd is a parallelogram
what is the value of n?

Explanation:

Step1: Use the property of a parallelogram

In a parallelogram, consecutive angles are supplementary. So, \((2n + 32)+(4n - 2)=180\).

Step2: Simplify the equation

Combine like - terms: \(2n+4n+32 - 2=180\), which gives \(6n+30 = 180\).

Step3: Solve for \(n\)

Subtract 30 from both sides: \(6n=180 - 30\), so \(6n=150\). Then divide both sides by 6: \(n=\frac{150}{6}=25\). Wait, there is a mistake. Let's check again.

Wait, another property: In a parallelogram, opposite angles are equal. But if we consider consecutive angles: \(A + B=180\) (if \(AB\parallel CD\)). But if we use the property that in a parallelogram \(AD\parallel BC\), then \(\angle A+\angle B = 180\) (consecutive interior angles). But actually, if we use the property that in a parallelogram \(\angle A=\angle C\) and \(\angle B=\angle D\), but if we consider the sum of adjacent angles. Wait, no, the correct property is that in a parallelogram \(AD\parallel BC\), so \(\angle A+\angle B = 180\) (consecutive interior angles). But if we assume that \(\angle B\) and \(\angle D\) are related. Wait, no, the correct formula is: In a parallelogram \(AD\parallel BC\), so \(\angle A+\angle B=180\). But if we use the property that \(\angle A\) and \(\angle D\) are related. Wait, no, the correct approach is: In a parallelogram \(AB\parallel CD\), so \(\angle B+\angle C = 180\) and \(\angle A+\angle D=180\). But if we use the property that \(\angle B\) and \(\angle D\) are consecutive angles (no, \(\angle B\) and \(\angle D\) are not consecutive. Wait, no, in a parallelogram \(ABCD\), \(AB\parallel CD\) and \(AD\parallel BC\). The sum of adjacent angles is \(180^{\circ}\). So \(\angle B+\angle A=180\), \(\angle A+\angle D = 180\), \(\angle D+\angle C=180\), \(\angle C+\angle B=180\). But if we assume that \(\angle B=(2n + 32)\) and \(\angle D=(4n-2)\) and since \(AB\parallel CD\), \(\angle B+\angle C=180\) and \(\angle A+\angle D=180\), but also \(\angle A=\angle C\) and \(\angle B=\angle D\) (opposite angles are equal). So \(2n + 32=4n-2\).

Step4: Solve the correct equation

Subtract \(2n\) from both sides: \(32=4n-2 - 2n\), so \(32 = 2n-2\). Add 2 to both sides: \(34=2n\). Divide by 2: \(n = 17\).

Answer:

\(17\)