14. In \( \triangle R S T \), if \( m \angle R \) is five more than twice \( x, m \angle S \) is one more than \( x \), and \( m \angle T \) is sixteen less than seven times \( x \), find \( x \) and the measure of each angle. \begin{tabular}{l}\( x= \) \\ \( m \angle R= \) \\ \( m \angle S= \) \\ \( m \angle T= \) \\ \hline\end{tabular}

To solve for \( x \) and the measures of each angle in triangle \( RST \), we'll first express each angle in terms of \( x \): - For angle \( R \): \( m \angle R = 2x + 5 \) - For angle \( S \): \( m \angle S = x + 1 \) - For angle \( T \): \( m \angle T = 7x - 16 \) According to the triangle angle sum property, the sum of the angles in a triangle is \( 180^\circ \): \[ m \angle R + m \angle S + m \angle T = 180^\circ \] Substituting the expressions we found: \[ (2x + 5) + (x + 1) + (7x - 16) = 180 \] Now, combine like terms: \[ 2x + x + 7x + 5 + 1 - 16 = 180 \] \[ 10x - 10 = 180 \] Next, add 10 to both sides: \[ 10x = 190 \] Now, divide by 10: \[ x = 19 \] Now that we have \( x \), we can find each angle: 1. For \( m \angle R \): \[ m \angle R = 2(19) + 5 = 38 + 5 = 43^\circ \] 2. For \( m \angle S \): \[ m \angle S = 19 + 1 = 20^\circ \] 3. For \( m \angle T \): \[ m \angle T = 7(19) - 16 = 133 - 16 = 117^\circ \] Thus, we can summarize the results as follows: \[ \begin{tabular}{l} \( x = 19 \) \\ \( m \angle R = 43^\circ \) \\ \( m \angle S = 20^\circ \) \\ \( m \angle T = 117^\circ \) \\ \hline \end{tabular} \]