A triangle has two congruent angles and one unique angle. The system of equations shown represents the relationship between the angle measures. What are the measures of the angles in the triangle? \( \left\{\begin{array}{l}2 x+y=180 \\ x-y=30\end{array}\right. \)

To solve for the angle measures in this triangle, let's interpret the equations. From the first equation, we have \(2x + y = 180\), where \(x\) represents the measure of each of the two congruent angles, and \(y\) is the unique angle. The second equation, \(x - y = 30\), tells us that the unique angle \(y\) is 30 degrees less than each of the congruent angles. Now, let's solve the system! Rearranging the second equation gives us \(y = x - 30\). We can substitute this into the first equation: \[ 2x + (x - 30) = 180 \] \[ 3x - 30 = 180 \] \[ 3x = 210 \] \[ x = 70 \] Now substituting \(x = 70\) back into \(y = x - 30\): \[ y = 70 - 30 = 40 \] Thus, the measures of the angles are \(70^\circ, 70^\circ, \text{ and } 40^\circ\). So, we have a triangle with two congruent angles measuring \(70\) degrees each and a unique angle measuring \(40\) degrees.