In ΔIJK, j = 460 cm, i = 440 cm and ∠I=69°. Find all possible values of ∠J, to the nearest degree.

To solve this problem, we will 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 the side of length c, the following equation holds:

c² = a² + b² - 2ab cos(γ)

In this case, we have a triangle IJK with sides i, j, and k, and we know the lengths of sides i and j, and the measure of angle I. We can rearrange the Law of Cosines to solve for the cosine of angle J:

cos(J) = (i² + j² - k²) / (2ij)

We don't know the length of side k, but we can find it using the Law of Cosines with the given information:

k² = i² + j² - 2ij cos(I)

Substituting the given values:

k² = (440 cm)² + (460 cm)² - 2(440 cm)(460 cm) cos(69°)

Calculate the value of k² and take the square root to find the length of side k. Then substitute the lengths of sides i, j, and k into the rearranged Law of Cosines to find the cosine of angle J. Finally, use the inverse cosine function to find the measure of angle J in degrees.