Given: ABCD – trapezoid, AD = 10cm, BC = 6cm, angle D = 45 degrees, angle A = 90 degrees. Find: СD.

From the top of the obtuse angle C we draw the height of the НС.

Quadrangle ABCН is a rectangle, since the sides BC and AH are parallel as the base of the trapezoid, AH belongs to AD, the sides AB and CH are parallel as the heights of the trapezoid, then AH = BC = 6 cm.

Let us determine the length of the DН segment. DН = AD – AН = 10 – 6 = 4 cm.

In a right-angled СDН triangle, the СDН angle = 450, then the DСН angle = 180 – 90 – 45 = 45, which means that the DСН triangle is isosceles, CH = DН = 4 cm.

Then, according to the Pythagorean theorem, the side of the СD will be equal to: СD^2 = CH^2 + DН^2 = 16 + 16 = 32.

СD = 4 * √2 cm.

Answer: The length of the side of the СD is 4 * √2 cm.