Question

using the supplementary consecutive angles theorem
manuela is making a banner for a school dance. she makes one angle $116^{circ}$. what should she make the measures of angles b and d so that the banner is a parallelogram?

Explanation:

Step1: Use the property of consecutive angles in a parallelogram

In a parallelogram, consecutive angles are supplementary. So, \(\angle A+\angle B = 180^{\circ}\), \(\angle B+\angle C=180^{\circ}\), \(\angle C+\angle D = 180^{\circ}\), \(\angle D+\angle A=180^{\circ}\). Given \(\angle C = 116^{\circ}\).
Since \(\angle B\) and \(\angle C\) are consecutive angles, \(\angle B=180^{\circ}-\angle C\).
\(\angle B = 180 - 116=64^{\circ}\)

Step2: Use the property of opposite angles in a parallelogram

In a parallelogram, opposite angles are equal. So, \(\angle D=\angle B\) (This is wrong. Wait, no, in a parallelogram \(\angle A=\angle C\) and \(\angle B=\angle D\). Wait, using the consecutive - angle property again. Since \(\angle D\) and \(\angle C\) are consecutive (if we consider side \(DC\) and \(AD\)), \(\angle D = 180^{\circ}-\angle C\) is wrong. Wait, no, \(\angle A\) and \(\angle D\) are consecutive ( \(AB\parallel DC\), \(AD\) is a transversal). Wait, no, the correct way:
In a parallelogram \(ABCD\) with \(AB\parallel DC\) and \(AD\parallel BC\).
We know that \(\angle B\) and \(\angle C\) are consecutive (\(AB\parallel DC\), \(BC\) is a transversal), \(\angle B = 180 - 116=64^{\circ}\)
Also, \(\angle D\) and \(\angle C\) are consecutive (\(AD\parallel BC\), \(DC\) is a transversal). So \(\angle D=180 - 116 = 64^{\circ}\) (Wait, no, wrong. Wait, in a parallelogram \(ABCD\), \(\angle A=\angle C\), \(\angle B=\angle D\). Using the consecutive - angle property: \(\angle A+\angle B=180\), \(\angle B+\angle C = 180\), \(\angle C+\angle D=180\), \(\angle D+\angle A=180\). Given \(\angle C = 116^{\circ}\), then \(\angle A=\angle C = 116^{\circ}\) (opposite angles). And \(\angle B=\angle D\) (opposite angles). Also, \(\angle B=180-\angle A\) (since \(AD\parallel BC\), \(AB\) is a transversal)
\(\angle B=180 - 116=64^{\circ}\), so \(\angle D=\angle B = 64^{\circ}\)

Answer:

\(\angle B = 64^{\circ}\) and \(\angle D=64^{\circ}\)