Question

find the measure of \\( \angle d \\), and the perimeter and area of parallelogram \\( a b c d \\).
\\( \angle d=60^{\circ}, p = 41 \\) centimeters, \\( a = 200 \\) square centimeters
\\( \angle d=150^{\circ}, p = 82 \\) centimeters, \\( a = 200 \\) square centimeters
\\( \angle d=60^{\circ}, p = 41 \\) centimeters, \\( a = 400 \\) square centimeters
\\( \angle d=160^{\circ}, p = 82 \\) centimeters, \\( a = 400 \\) square centimeters

Explanation:

Step1: Find the measure of ∠D

In a parallelogram, consecutive angles are supplementary. So, ∠A + ∠D = 180°. Given ∠A = 30°, then ∠D = 180° - 30° = 150°.

Step2: Calculate the perimeter of parallelogram ABCD

The formula for the perimeter \(P\) of a parallelogram is \(P = 2(a + b)\), where \(a = 25\) cm and \(b = 16\) cm. So, \(P=2(25 + 16)=2\times41 = 82\) cm.

Step3: Calculate the area of parallelogram ABCD

The formula for the area \(A\) of a parallelogram is \(A=\text{base}\times\text{height}\). Here, base \(= 25\) cm and height \( = 8\) cm. So, \(A=25\times8=200\) \(cm^{2}\).

Answer:

∠D = 150°, \(P = 82\) centimeters, \(A = 200\) square centimeters (the second option).