Question

perimeter = 164. solve for x and y. x = 81° y y + 12 cm y - 12 cm x

Explanation:

Step1: Use the property of parallelogram

In a parallelogram, opposite sides are equal. So, \(y=y\), \(y + 12=y-12\) (This is wrong, actually, for the perimeter formula of a parallelogram \(P = 2(a + b)\), where \(a\) and \(b\) are adjacent sides. Here, adjacent sides are \(y\) and \(y + 12\))
The perimeter formula \(P=2(y+(y + 12))\). Given \(P = 164\).
So, \(164=2(y+(y + 12))\)
Divide both sides by 2: \(82=y+(y + 12)\)
Simplify: \(82=2y+12\)
Subtract 12 from both sides: \(2y=82 - 12=70\)
Divide by 2: \(y = 35\)

Step2: Use the angle - side relationship

In a parallelogram, consecutive angles are supplementary. One angle is \(81^{\circ}\), and the angle adjacent to it (where \(x\) is involved) satisfies \(x+81^{\circ}=180^{\circ}\)
Solve for \(x\): \(x=180^{\circ}-81^{\circ}=99^{\circ}\)

Answer:

\(x = 99^{\circ}\), \(y = 35\)