In quadrilateral ABCD above , `bar(AD) bar(BC)` and `CD=1/2AB`. What is the measure of angle B ?

Options

A. `150^@`
B. `135^@`
C. `120^@`
D. `90^@`

Answer

A

Explanation

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 rectangle FBCD such that FB = CD. This will also create right triangle ABF such that `FB = 1/2AB`. An acute angle in a right triangle has measure `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=1/2AB`, 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^@`.
Choice B is incorrect and may result from identifying triangle ABF as a `45^@-45^@-90^@` triangle and the measure of angle ABF as `45^@`. Choice C is incorrect and may result from adding the measures of angles BAF and FBC rather than angles ABF and FBC. Choice D is incorrect and may result from finding the measure of angle D rather than angle B.