In the figure above, lines L and P are parallel. The segment AD is the same length as the segment DC; AD is parallel to BC. If the length of AD is equal to 4, and angle ADC is equal to 60º, what is the area of ABCD?

A. 4√2

B. 4√3

C. 8√2

D. 8√3

E. 16√2

(D) AB is parallel to DC and AD is parallel to BC, so ABCD is parallelogram. We know how to find the area of a parallelogram we multiply the height of parallelogram by its base length.

We are told that the angle ADC is a 60º angle. If we draw an altitude from A straight down and perpendicular to line P, the length of that altitude will be equal to the height of the rhombus. Moreover, it forms a 60-30-90 triangle, so we can easily find its length. Since the hypotenuse of that triangle is equal to 4, the second longest side must be equal to 2√3 (since the proportions for the triangle run

*x*,

*x*√3, and 2

*x*).

To find the area of a rhombus, multiply the height of the rhombus by its base length: 4 × 2√3 = 8√3, or answer choice (A).

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**The sum of the interior angles should be (***n* – 2) × 180. So the sum of the interior angles for ABCD = 360⁰. How is it determined that the traingle formed from A is a 30-60-90 triangle?