In quadrilateral ABCD, AD  BC and CD = 1/2 AB. What is the measure of angle B?

Explanation

Choice A is correct. A segment can be drawn inside of quadrilateral ABCD from point B to point F (not shown) on segment AD such that segment BF is perpendicular to segment AD. This will create a rectangle FBCD such that FB = CD. This will also create a right triangle ABF such that FB = 1/2 AB. An acute angle in a right triangle measures 30° if and only if the side opposite this angle is half the length of the hypotenuse. (Such a triangle is called a 30°-60°-90° triangle.) Since AB is the hypotenuse of right triangle ABF and FB = 1/2 AB, triangle ABF must be a 30°-60°-90° triangle, and angle ABF must measure 60°. The measure of angle ABC equals the sum of the measures of angles ABF and FBC. Because angle FBC is in rectangle FBCD, it has a measure of 90°. Therefore, the measure of angle ABC, or angle B shown in the original figure, is 60° + 90° = 150°.