In triangle XYZ, angle X measures 45 degrees, side y is 8 units, and side z is 6 units. Use the Law of Cosines to find the measure of angle Y.

First, let's apply the Law of Cosines, which states that \( c^2 = a^2 + b^2 - 2ab \cdot \cos(C) \). In triangle XYZ, we want to find angle Y, so we can rearrange this formula as follows: \[ y^2 = x^2 + z^2 - 2xz \cdot \cos(Y) \] Here, we know \( y = 8 \), \( z = 6 \), and angle \( X = 45^\circ \) (which we need to find out its opposite side, \( x \)). Now, using the Law of Sines to find side \( x \): \[ \frac{y}{\sin(Y)} = \frac{z}{\sin(X)} \] Substituting the values we know: \[ \frac{8}{\sin(Y)} = \frac{6}{\sin(45^\circ)} \] Since \( \sin(45^\circ) = \frac{\sqrt{2}}{2} \), rearranging gives us: \[ \sin(Y) = \frac{8 \cdot \sin(45^\circ)}{6} = \frac{8 \cdot \frac{\sqrt{2}}{2}}{6} = \frac{4\sqrt{2}}{6} = \frac{2\sqrt{2}}{3} \] Now to find angle Y, we take the arcsin: \[ Y = \arcsin\left(\frac{2\sqrt{2}}{3}\right) \] Using a calculator gives \( Y \approx 41.81^\circ \). Now, let's use this measure to find angle Z with the triangle angle sum property: \[ Z = 180^\circ - X - Y = 180^\circ - 45^\circ - 41.81^\circ \approx 93.19^\circ \] So the measure of angle Y in triangle XYZ is approximately \( 41.81^\circ \).