In ΔUVW, with sides u, w, and angle ∠W known, two possible values for ∠U exist using the Law of Sines:
Acute Angle U: 65.8° (valid solution)
Obtuse Angle U: 114.2° (invalid due to exceeding 180°)
Therefore, ∠U ≈ 65.8° (to the nearest 10th).
In triangle ΔUVW, with sides u = 5.9 cm, w = 6.2 cm, and angle ∠W = 74°, we can determine two possible values for ∠U using the Law of Sines:
Acute Angle U:
Using the formula sin(∠U)/u = sin(∠W)/w, we get sin(∠U) = (sin(74°)*5.9)/6.2 ≈ 0.908. Solving for the angle, we find two solutions: ∠U ≈ 65.8° (acute) and ∠U ≈ 180° - 65.8° ≈ 114.2° (acute).
Obtuse Angle U:
However, the Law of Sines only guarantees solutions within the range 0° to 180°. Since the sum of angles in a triangle is 180°, the third angle (∠V) must be: ∠V = 180° - (∠W + ∠U)
Therefore, for the acute case (∠U ≈ 65.8°), ∠V ≈ 40.4° and for the obtuse case (∠U ≈ 114.2°), ∠V ≈ 65.8°.
Since angles cannot exceed 180°, the valid solution for ∠U is the acute case: ∠U ≈ 65.8° (to the nearest 10th of a degree).
This solution satisfies all triangle properties and aligns with the given angle measures.