In▲ABC, if AC = 12, BC = 11, and measure of angle A = 30 degrees, angle C could be
(1) an obtuse angle, only
(2) an acute angle, only
(3) a right angle, only
(4) either an obtuse angle or an acute angle
Draw a picture! Draw line segment AC with length 12, angle A of 30 degrees- that is draw a ray without end starting at A and at 30 degrees. At the other end you know CB has length 11 but you don't know its direction so use compasses to strike an arc with radius 11. Where does it cross the ray you drew?
There are three possiblities- if the radius were too short it wouldn't cross at all and there would be no triangle at all. With "11" and "12" that doesn't happen here. If this happened to be a right triangle, it would be just tangent to the ray. That doesn't happen here- there is no right triangle with angle 30, opposite leg 11 and hypotenuse 12: sin(30) is NOT 11/12.
What happens here is that the arc of radius 11 crosses that ray twice. There are two possible triangles, one with an acute angle at B, the other with an obtuse angle at B.