In \( \Delta \mathrm{JKL}, j=820 \) inches, \( k=530 \) inches and \( l=960 \) inches. Find the measure of \( \angle \mathrm{K} \) to the nearest degree.

To find the measure of \( \angle K \) in \( \Delta JKL \), we can use the Law of Cosines, which states that for any triangle with sides \( a \), \( b \), and \( c \), and the corresponding opposite angles \( A \), \( B \), and \( C \): \[ c^2 = a^2 + b^2 - 2ab \cdot \cos(C) \] In this case, let \( j = 820 \) inches, \( k = 530 \) inches, and \( l = 960 \) inches. We are trying to find \( \angle K \), which is opposite side \( l \). Rearranging the Law of Cosines for \( \angle K \): \[ l^2 = j^2 + k^2 - 2jk \cdot \cos(K) \] Substituting the values we have: \[ 960^2 = 820^2 + 530^2 - 2 \cdot 820 \cdot 530 \cdot \cos(K) \] Calculating: \[ 921600 = 672400 + 280900 - 2 \cdot 820 \cdot 530 \cdot \cos(K) \] \[ 921600 = 953300 - 871200 \cdot \cos(K) \] Rearranging gives: \[ 871200 \cdot \cos(K) = 953300 - 921600 \] \[ 871200 \cdot \cos(K) = 31600 \] Dividing by 871200: \[ \cos(K) = \frac{31600}{871200} \approx 0.0362 \] Now find \( \angle K \): \[ K \approx \cos^{-1}(0.0362) \] Using a calculator: \[ K \approx 88.9^\circ \] Rounding to the nearest degree, we get: \[ \angle K \approx 89^\circ \]