Find the largest angle of an isosceles trapezoid ABCD if the diagonal AC makes

Find the largest angle of an isosceles trapezoid ABCD if the diagonal AC makes angles equal to 30 and 45 degrees with the base AD and the lateral side AB, respectively.

Determine the value of the angle BAD.

Angle BAD = BAC + DAC = 45 + 30 = 75.

Since, by condition, the trapezoid is isosceles, then its angles at the base are equal, then the angle CAD = BAD = 75.

The sum of the angles of the trapezoid with its lateral side is 180, then the angle ABC = DCB = (180 – BAD) = (180 – 75) = 105.

Answer: The largest angle of the trapezoid is 105.