24. Step 1: Calculate the determinant of the given $3 \times 3$ matrix. The determinant of a matrix $\begin{vmatrix} a & b & c \\ d & e & f \\ g & h & i \end{vmatrix}$ can be found by expanding along the first column: $a(ei - fh) - d(bi - ch) + g(bf - ce)$. For the given matrix $\begin{vmatrix} 0 & 3 & 2 \\ 1 & 7 & 8 \\ 0 & 5 & 4 \end{vmatrix}$, we expand along the first column: $$ \det = 0 \cdot \begin{vmatrix} 7 & 8 \\ 5 & 4 \end{vmatrix} - 1 \cdot \begin{vmatrix} 3 & 2 \\ 5 & 4 \end{vmatrix} + 0 \cdot \begin{vmatrix} 3 & 2 \\ 7 & 8 \end{vmatrix} $$ Step 2: Simplify the expression. $$ \det = 0 - 1 \cdot ((3 \times 4) - (2 \times 5)) + 0 $$ $$ \det = -1 \cdot (12 - 10) $$ $$ \det = -1 \cdot (2) $$ $$ \det = -2 $$ The correct option is A. The final answer is $\boxed{\text{A. -2}}$. 25. Step 1: Find the exterior angle of the regular polygon. The interior angle of the regular polygon is given as $135^\circ$. The sum of an interior angle and its corresponding exterior angle is $180^\circ$. $$ \text{Exterior Angle} = 180^\circ - \text{Interior Angle} $$ $$ \text{Exterior Angle} = 180^\circ - 135^\circ $$ $$ \text{Exterior Angle} = 45^\circ $$ Step 2: Calculate the number of sides of the polygon. The sum of the exterior angles of any polygon is $360^\circ$. For a regular polygon with $n$ sides, all exterior angles are equal. $$ n \times \text{Exterior Angle} = 360^\circ $$ $$ n \times 45^\circ = 360^\circ $$ $$ n = \frac{360^\circ}{45^\circ} $$ $$ n = 8 $$ The correct option is A. The final answer is $\boxed{\text{A. 8}}$. 26. Step 1: Use the property of parallel lines to find $\angle LNM$. Given that $KL \parallel NM$ and $LN$ is a transversal, the alternate interior angles are equal. Therefore, $\angle KLN = \angle LNM$. Given $\angle KLN = 54^\circ$, so $\angle LNM = 54^\circ$. Step 2: Use the angle bisector property to find $\angle KNL$. Given that $LN$ bisects $\angle KNM$, it means $LN$ divides $\angle KNM$ into two equal angles. So, $\angle KNL = \angle LNM$. Since $\angle LNM = 54^\circ$, then $\angle KNL = 54^\circ$. Step 3: Calculate the total angle $\angle KNM$. $$ \angle KNM = \angle KNL + \angle LNM $$ $$ \angle KNM = 54^\circ + 54^\circ $$ $$ \angle KNM = 108^\circ $$ Step 4: Calculate $\angle KMN$ using the sum of angles in $\triangle KMN$. The sum of angles in a triangle is $180^\circ$. In $\triangle KMN$: $$ \angle KMN + \angle MKN + \angle KNM = 180^\circ $$ Given $\angle MKN = 35^\circ$ and we found $\angle KNM = 108^\circ$. $$ \angle KMN + 35^\circ + 108^\circ = 180^\circ $$ $$ \angle KMN + 143^\circ = 180^\circ $$ $$ \angle KMN = 180^\circ - 143^\circ $$ $$ \angle KMN = 37^\circ $$ The final answer is $\boxed{37^\circ}$. Send me the next one 📸