2. In \( \triangle X Y Z_{z}, X Z=17, X Y=18 \), and \( Y Z=14 \). Find \( m \angle Y \).

To find \( m \angle Y \) in triangle \( XYZ \) where \( XZ = 17 \), \( XY = 18 \), and \( YZ = 14 \), we can use the Law of Cosines. The Law of Cosines states: \[ c^2 = a^2 + b^2 - 2ab \cdot \cos(C) \] In our triangle, let \( a = YZ = 14 \), \( b = XZ = 17 \), and \( c = XY = 18 \). Since we need to find \( m \angle Y \), we substitute into the formula as follows: \[ 18^2 = 14^2 + 17^2 - 2 \cdot 14 \cdot 17 \cdot \cos(Y) \] Calculating the squares: \[ 324 = 196 + 289 - 2 \cdot 14 \cdot 17 \cdot \cos(Y) \] Calculating \( 196 + 289 \): \[ 324 = 485 - 2 \cdot 14 \cdot 17 \cdot \cos(Y) \] Now, simplifying: \[ 324 - 485 = - 2 \cdot 14 \cdot 17 \cdot \cos(Y) \] \[ -161 = - 2 \cdot 14 \cdot 17 \cdot \cos(Y) \] Calculating \( 2 \cdot 14 \cdot 17 \): \[ 2 \cdot 14 = 28, \quad 28 \cdot 17 = 476 \] So we have: \[ -161 = -476 \cdot \cos(Y) \] Dividing both sides by \(-476\): \[ \cos(Y) = \frac{161}{476} \] Now calculating \( \cos^{-1}(Y) \): \[ m \angle Y = \cos^{-1}\left(\frac{161}{476}\right) \] Using a calculator, we find: \[ m \angle Y \approx 63.7^\circ \] Therefore, \( m \angle Y \approx 63.7^\circ \).