14. In \( \Delta R S T \), 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 \( \Delta RST \), we start with the information given: 1. \( m \angle R = 2x + 5 \) 2. \( m \angle S = x + 1 \) 3. \( m \angle T = 7x - 16 \) Using the triangle 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 in our expressions for the angles, we have: \[ (2x + 5) + (x + 1) + (7x - 16) = 180 \] Now, combine like terms: \[ 2x + x + 7x + 5 + 1 - 16 = 180 \] This simplifies to: \[ 10x - 10 = 180 \] Now, add \( 10 \) to both sides: \[ 10x = 190 \] Next, we divide by \( 10 \): \[ x = 19 \] Now we can find the measures of the angles by substituting \( x \) back into the expressions: 1. \( m \angle R = 2(19) + 5 = 38 + 5 = 43^\circ \) 2. \( m \angle S = 19 + 1 = 20^\circ \) 3. \( m \angle T = 7(19) - 16 = 133 - 16 = 117^\circ \) Now we can present our final results: \[ \begin{tabular}{l} \( x = 19 \) \\ \( m \angle R = 43^\circ \) \\ \( m \angle S = 20^\circ \) \\ \( m \angle T = 117^\circ \) \\ \hline \end{tabular} \]