### Instruction: One angle of a parallelogram is 120 degrees, and two consecutive sides have lengths of 8 inches and 15 inches. What is the area of the parallelogram? Express your answer in simplest radical form.### Response: To find the area of a parallelogram, I need to multiply the base by the height.
I can choose any side as the base, so I'll pick the 15-inch side.
The height is the perpendicular distance from the base to the opposite side, so I need to draw a perpendicular line from one of the vertices of the 15-inch side to the opposite side.
This line will form a right triangle with the 8-inch side and part of the 15-inch side.
I can use the angle of 120 degrees to find the measure of the angle opposite to the 8-inch side in the triangle.
Since the opposite angles of a parallelogram are congruent, the angle adjacent to the 120-degree angle is also 120 degrees.
That means the angle opposite to the 8-inch side in the triangle is 180 - 120 = 60 degrees.
Now I can use trigonometry to find the height of the parallelogram.
I know that the sine of 60 degrees is equal to the opposite side over the hypotenuse, so I can write sin(60) = h/8, where h is the height.
Solving for h, I get h = 8sin(60).
I recall that sin(60) is equal to the square root of 3 over 2, so I can simplify h to 4 times the square root of 3.
Now I have the base and the height of the parallelogram, so I can find the area by multiplying them.
The area is 15 times 4 times the square root of 3, which I can write as 60 times the square root of 3.