9.2. In a trapezoid, one lateral side is twice as large as the other, and the sum of the angles at the larger base is 120 degrees. Find the angles of the trapezoid.
Answer: 90 and 30 degrees.

Solution: Let the vertices of the trapezoid be A, B, C, D, with the larger base AD, and assume CD is twice as long as AB. Choose a point E on AD such that BE is parallel (and equal) to CD, and let M be the midpoint of segment BE. Then triangle ABM is isosceles with a 60-degree angle at vertex B, so it is equilateral, and point M is equidistant from A, B, and E. Therefore, M is the center of the circle containing points A, B, and E, and BE is its diameter. Angle BAE is an inscribed angle subtending this diameter, so its measure is 90 degrees, which means the measure of angle BAD is also 90 degrees, and the measure of angle CDA is 30 degrees.

Criteria: The idea of constructing segment BE: 2 points.