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. \[ \begin{array}{l}m \angle R= \\ m \angle S= \\ m \angle T=\end{array} \]

To solve the problem, let's first express the measures of angles R, S, and T in terms of \( x \): - \( m \angle R = 2x + 5 \) - \( m \angle S = x + 1 \) - \( m \angle T = 7x - 16 \) According to the triangle angle sum property, the sum of the angles in a triangle is \( 180^\circ \): \[ m \angle R + m \angle S + m \angle T = 180 \] Substituting the expressions for the angles: \[ (2x + 5) + (x + 1) + (7x - 16) = 180 \] Now, combine like terms: \[ 2x + 5 + x + 1 + 7x - 16 = 180 \] \[ (2x + x + 7x) + (5 + 1 - 16) = 180 \] \[ 10x - 10 = 180 \] Now, add 10 to both sides: \[ 10x = 190 \] Dividing both sides by 10 gives: \[ x = 19 \] Now we can find the measure of each angle: \[ m \angle R = 2(19) + 5 = 38 + 5 = 43 \] \[ m \angle S = 19 + 1 = 20 \] \[ m \angle T = 7(19) - 16 = 133 - 16 = 117 \] So, the measures of the angles are: \[ \begin{array}{l} m \angle R = 43^\circ \\ m \angle S = 20^\circ \\ m \angle T = 117^\circ \end{array} \]