Final answer:
To find ∠R, we subtract ∠Q (116°) from 180° and consider the sum of ∠R and ∠S, but without additional information, we cannot uniquely determine ∠R.
Step-by-step explanation:
To find ∠R in ΔQRS, where r = 89 inches, s = 32 inches, and ∠Q = 116°, we can use the fact that the sum of the angles in any triangle equals 180°. Given ∠Q, we can calculate as follows:
∠R + ∠S + ∠Q = 180°
∠R + ∠S + 116° = 180°
∠R + ∠S = 180° - 116°
∠R + ∠S = 64°
Since we do not have the measure of ∠S, but we know the sides r and s, we could apply the Law of Sines to find the ratio between the sides and the sines of their opposite angles. However, since we only need to find ∠R and we don't have the length of side q, we cannot use the Law of Sines here. Instead, we will use the fact that the angles ∠R and ∠S must also add up to 64°. Without further information, ∠R cannot be uniquely determined. Typically, if we had all three sides or two sides and an angle not in between those sides (SSA condition), we could proceed further in our calculations.