Question

15 parallelogram abcd is shown. what are the values of x and y? enter the correct values in the boxes. x = y =

Explanation:

Step1: Use the property of adjacent angles in a parallelogram

In a parallelogram, adjacent angles are supplementary (sum to \(180^{\circ}\)).
For \(\angle D\) and \(\angle C\), we have \(y + 70=180\).
Solving for \(y\):
\(y=180 - 70\)
\(y = 110\)

Step2: Use the property of opposite angles in a parallelogram

In a parallelogram, opposite angles are equal.
\(\angle B=x\) and \(\angle D = 70^{\circ}\), \(\angle B\) and \(\angle D\) are not opposite. But \(\angle A\) and \(\angle C\) are equal (\(110^{\circ}\)), \(\angle B\) and \(\angle D\) are supplementary.
Another way: Since \(AB\parallel CD\), \(\angle A+\angle D=180^{\circ}\), \(\angle A = 110^{\circ}\), and \(\angle B=\angle D\) (alternate - interior angles for \(AD\parallel BC\) and transversal \(AB\) is wrong. Correct: In parallelogram \(ABCD\), \(AD\parallel BC\), so \(\angle A+\angle B = 180^{\circ}\), \(\angle D+\angle C=180^{\circ}\), and \(\angle A=\angle C\), \(\angle B=\angle D\) (opposite angles). Wait, no. The correct property: In parallelogram \(ABCD\), \(AD\parallel BC\), so \(\angle A+\angle B=180^{\circ}\), \(AB\parallel CD\), so \(\angle B+\angle C = 180^{\circ}\), \(\angle A=\angle C\), \(\angle B=\angle D\).
Since \(AD\parallel BC\) and \(AB\) is a transversal, \(\angle A+\angle B=180^{\circ}\). Also, \(\angle D = 70^{\circ}\), and \(\angle B\) and \(\angle D\) are supplementary to \(\angle A\) and \(\angle C\) respectively. But the correct rule is: In parallelogram \(ABCD\), \(AB\parallel CD\) and \(AD\parallel BC\). \(\angle B\) and \(\angle D\) are supplementary to \(\angle A\) and \(\angle C\). But the most straightforward: \(AD\parallel BC\), so \(\angle A+\angle B=180^{\circ}\), \(AB\parallel CD\), so \(\angle D+\angle A=180^{\circ}\), then \(\angle B=\angle D = 70^{\circ}\) (alternate - interior angles for \(AB\parallel CD\) and transversal \(AD\) is wrong. Correct: In parallelogram \(ABCD\), \(\angle B=\angle D\) (opposite angles). Wait, no! The property is: In a parallelogram, opposite angles are equal and consecutive angles are supplementary. \(\angle D = 70^{\circ}\), \(\angle B\) and \(\angle D\) are opposite angles? No! \(\angle A\) and \(\angle C\) are opposite, \(\angle B\) and \(\angle D\) are opposite.
\(x = 110\), \(y=70\)

Answer:

\(x = 110\), \(y = 70\)