solve this---**Question Number:** 30
**Chart/Diagram Description:**
* Type: Geometric figure (quadrilateral).
* Main Elements:
* Points: A, B, C, D labeled as vertices of a quadrilateral.
* Lines: Segments AB, BC, CD, and DA forming the sides of the quadrilateral.
* Angles: A right angle symbol is shown at vertex C. Angle B is an obtuse angle. Angle A is an acute angle. Angle D appears to be a right angle based on the parallelism and the right angle at C, forming a trapezoid with a right angle.
* Relative Position: AD is shown as a horizontal base. BC is above AD and parallel to it. CD is a vertical segment connecting BC to AD, perpendicular to both (due to the right angle at C and parallelism). AB is a slanted segment connecting A to B.
* Relationships: AD is parallel to BC (indicated by text). CD is perpendicular to BC and AD (implied by the right angle at C and parallelism).
**Question Stem:**
In quadrilateral ABCD above, AD || BC and CD = (1/2) AB. What is the measure of angle B?
**Options:**
A) 150°
B) 135°
C) 120°
D) 90°
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In this problem, we have a quadrilateral ABCD where AD is parallel to BC, and CD equals one-half of AB. We need to find the measure of angle B. Let's first understand the shape. We can see that angles C and D are right angles, making this a right trapezoid with AD parallel to BC. The key relationship we need to use is that CD equals one-half of AB.
To solve this problem, let's draw a perpendicular line from point B down to line AD. This creates point E on AD. Notice that BCDE forms a rectangle, since BC is parallel to AD, and angles C and D are right angles. This means angle BED is also a right angle. Additionally, ABE forms a right triangle with the right angle at E. This decomposition will help us find angle B.
Now, let's use the given relationship that CD equals one-half of AB. Since BCDE is a rectangle, CD equals BE, as they are opposite sides of the rectangle. Therefore, BE also equals one-half of AB. Looking at the right triangle ABE, BE is the side opposite to angle A, and AB is the hypotenuse. The ratio of BE to AB is one-half, which means sine of angle A equals one-half. Therefore, angle A equals 30 degrees.
Now that we know angle A is 30 degrees, we can find angle ABE in the right triangle ABE. Since the sum of angles in a triangle is 180 degrees, and angle E is 90 degrees, angle ABE equals 180 minus 90 minus 30, which is 60 degrees. Next, we need to find angle B in the quadrilateral. Angle B is composed of two parts: angle ABE, which we just found to be 60 degrees, and angle CBE, which is 90 degrees because BCDE is a rectangle. Therefore, angle B equals 60 degrees plus 90 degrees, which gives us 150 degrees.
Let's summarize our solution. We identified that ABCD is a right trapezoid with AD parallel to BC. We drew a perpendicular line from B to AD, creating point E, and recognized that BCDE forms a rectangle. Since CD equals BE in the rectangle, and we're given that CD equals one-half of AB, we concluded that BE equals one-half of AB. In the right triangle ABE, sine of angle A equals BE divided by AB, which is one-half. Therefore, angle A equals 30 degrees. In triangle ABE, angle ABE equals 180 minus 90 minus 30, which is 60 degrees. Finally, angle B equals angle ABE plus angle CBE, which is 60 degrees plus 90 degrees, giving us 150 degrees. Therefore, the answer is option A: 150 degrees.