Answer:
A. single triangle. C = 8.74°, A = 1.26°, a = 1.01
Step-by-step explanation:
When two sides and an angle are given, the possibility of two (or zero) triangles exists only when the given angle is opposite the shorter of the two given sides. That is not the case here, so the given measures define one unique triangle.
Law of Sines
The law of sines tells us the sides and angles have the relation ...
sin(A)/a = sin(B)/b = sin(C)/c
We can use this first to find angle C:
sin(C) = c/b·sin(B)
C = arcsin(c/b·sin(B)) = arcsin(7/8·sin(170°)) ≈ 8.7394°
∠C ≈ 8.74°
Remaining measures
Now that we know two angles, we can find the third:
A = 180° -B -C = 180° -170° -8.74° = 1.26°
∠A = 1.26°
Using the law of sines again, we can find the measure of side 'a'.
a = b·sin(A)/sin(B) = 8·sin(1.26°)/sin(170°) ≈ 1.01346
The measure of segment 'a' is about 1.01 units.
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Additional comment
If 'a', 'b', and angle A are given, there will be zero triangles if b/a·sin(A) > 1. If b/a·sin(A) < 1 and b > a, there will be two (2) triangles. Otherwise, as here, there will be one unique triangle.
