In \( \triangle Q R S \), if \( \overline{Q R} \cong \overline{R S}, m \angle Q=(8 x-17)^{\circ}, m \angle R=(19 x+4)^{\circ} \). and \( m \angle S=(5 x+1)^{\circ} \). find the value of \( x \) and the measure of each angle. \( M

In triangle \( \triangle QRS \), we know that \( \overline{QR} \cong \overline{RS} \), which means that triangle \( QRS \) is isosceles with \( \angle Q \) and \( \angle S \) being the base angles. Therefore, we have: \[ m \angle Q = m \angle S \] Given the measures of the angles: \[ m \angle Q = (8x - 17)^\circ \] \[ m \angle S = (5x + 1)^\circ \] Setting these two expressions equal to each other, we get: \[ 8x - 17 = 5x + 1 \] Now, we can solve for \( x \): 1. Subtract \( 5x \) from both sides: \[ 8x - 5x - 17 = 1 \] \[ 3x - 17 = 1 \] 2. Add 17 to both sides: \[ 3x = 18 \] 3. Divide by 3: \[ x = 6 \] Now that we have \( x \), we can find the measures of the angles: 1. Calculate \( m \angle Q \): \[ m \angle Q = 8x - 17 = 8(6) - 17 = 48 - 17 = 31^\circ \] 2. Calculate \( m \angle S \): \[ m \angle S = 5x + 1 = 5(6) + 1 = 30 + 1 = 31^\circ \] 3. Calculate \( m \angle R \): \[ m \angle R = 19x + 4 = 19(6) + 4 = 114 + 4 = 118^\circ \] Now we have the measures of all angles: - \( m \angle Q = 31^\circ \) - \( m \angle R = 118^\circ \) - \( m \angle S = 31^\circ \) Finally, we can summarize the results: - The value of \( x \) is \( 6 \). - The measures of the angles are: - \( m \angle Q = 31^\circ \) - \( m \angle R = 118^\circ \) - \( m \angle S = 31^\circ \) Thus, the measure of angle \( Q \) is: \[ \boxed{31^\circ} \]