In ΔWXY, y = 70 inches, x = 66 inches and ∠X=161°. Find all possible values of ∠Y, to the nearest degree.

To find the value of ∠Y in ΔWXY, we can use the Law of Cosines. The Law of Cosines states that for any triangle with sides of lengths a, b, and c and an angle γ opposite side c, the following equation holds true:

c² = a² + b² - 2ab cos(γ)

In this case, we know that side x = 66 inches, side y = 70 inches, and ∠X = 161°. We want to find ∠Y, the angle opposite side y. So we can set up the equation as follows:

y² = x² + w² - 2xw cos(∠X)

70² = 66² + w² - 266w cos(161°)

However, we don't know the length of side w, so we can't solve this equation directly. Instead, we can use the Law of Sines to find ∠Y. The Law of Sines states that the ratio of the length of a side of a triangle to the sine of the angle opposite that side is the same for all three sides of the triangle. So we can set up the following equation:

x/sin(∠X) = y/sin(∠Y)

66/sin(161°) = 70/sin(∠Y)

Solving this equation for ∠Y gives us:

∠Y = arcsin[(70*sin(161°))/66]

Calculating this gives us ∠Y ≈ 19° or ∠Y ≈ 161°. However, in a triangle, the sum of all angles must be 180°. Given that ∠X = 161°, ∠Y cannot be 161° because that would make the sum of the angles greater than 180°. Therefore, ∠Y must be 19°.