In trapezium ABCD AB and CD base, AB = BC = 8, angle C = angle D = 60 degrees. Find the area of the trapezoid.

We will construct the heights АН and ВК of the trapezoid.

Since the angles at the base of the СD are equal, the trapezoid of AВСD is isosceles, Then AD = AB = BC = 8 cm.

The triangles AНD and ВCК are rectangular in which the angles DAН = СВK = (90 – 60) = 30.

Then the segments DН = СK = BC / 2 = 8/2 = 4 cm.

Quadrangle AВKН is a rectangle, then НK = AB = 8 cm.

Then the length of the segment AC = DН + НK + СK = 4 + 8 + 4 = 16 cm.

In a right-angled triangle AВD, according to the Pythagorean theorem, AH^2 = AD^2 – DN^2 = 64 – 16 = 48.

AH = 4 * √3 cm.

Then Savsd = (AB + СD) * AН / 2 = (8 + 16) * 4 * √3 / 2 = 48 * √3 cm.

Answer: The area of the trapezoid is 48 * √3 cm.