Answer:
A) b > 10
B) none
C) b ≤ 10
Step-by-step explanation:
For the purpose of this answer, we assume ABC is a triangle if and only if all angles and side lengths have measures greater than 0, and the sum of angles is exactly 180°.
The fact that B = 150° means A+C = 30°. That is, 0 < A < 30°. Taking the sine of A, we have ...
0 < sin(A) < 1/2
A, B)
The Law of Sines tells us ...
sin(A)/a = sin(B)/b
sin(A) = a·sin(B)/b = 10·(1/2)/b = 5/b
Substituting this into the inequality for sin(A), we get ...
0 < 5/b < 1/2
0 < 10 < b . . . . . multiply by 2b (which is > 0)
In order to form a triangle, the value of b must be greater than 10. In that case, there will always be only one value for A:
A = arcsin(5/b) . . . . b > 10
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C)
For b = 10, the "triangle" is a straight line, a disallowed condition. That is A will have no allowed value.
For b < 10, A > 30°, which forces C < 0°. This is also a disallowed condition. So we can say ...
b ≤ 10, A has no allowed value
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The attached diagram shows triangle ABC. Point A can be anywhere on ray BA. As segment c gets longer, angle C increases from near 0 to near 30°. At the same time, angle A decreases from near 30° to near 0°. When b=10, c=0, and the figure is not a triangle. Nor is it a triangle for b < 10.
