In isosceles triangle \( A B C, \angle B \) is the vertex angle. \( \angle A=5 x+8 \angle C=7 x-10 \). What are the measures of each base angle? \( 99^{\circ} \) \( 74^{\circ} \) \( 60^{\circ} \) \( 53^{\circ} \)

To find the measures of the base angles in triangle \( ABC \), we start by recognizing that the sum of the angles in any triangle is \( 180^\circ \). Since \( ABC \) is an isosceles triangle, the two base angles \( A \) and \( C \) are equal. Thus, we can set the equations for the angles as follows: \[ \angle A = \angle C \] From the problem, we have: \[ 5x + 8 = 7x - 10 \] Solving for \( x \): 1. Rearranging the equation: \[ 5x + 8 - 7x + 10 = 0 \] \[ -2x + 18 = 0 \] \[ 2x = 18 \] \[ x = 9 \] Now we can find the actual measures of the angles: \[ \angle A = 5(9) + 8 = 45 + 8 = 53^\circ \] \[ \angle C = 7(9) - 10 = 63 - 10 = 53^\circ \] Given that the triangle is isosceles, \( \angle B \) can be calculated as follows: \[ \angle B = 180^\circ - \angle A - \angle C = 180^\circ - 53^\circ - 53^\circ = 74^\circ \] Thus, the measures of the base angles \( A \) and \( C \) are \( 53^\circ \).