One of the angles of a triangle is two-third of the sum of the adjacent angles of a parallelogram. The remaining angles of the triangle are in the ratio of 5 : 7. What is the value of the second largest angle of the triangle?

Let’s denote the angles of the triangle as \(A\), \(B\), and \(C\), where \(A\) is the angle that is two-third of the sum of the adjacent angles of a parallelogram, and \(B\) and \(C\) are the remaining angles of the triangle in the ratio of 5:7.

Since the sum of the angles in a triangle is \(180^\circ\), we can write:

\[ A + B + C = 180^\circ \]

Given that \(A\) is two-third of the sum of the adjacent angles of a parallelogram, and we know that the sum of the adjacent angles of a parallelogram is \(180^\circ\), we have:

\[ A = \frac{2}{3} \times 180^\circ = 120^\circ \]

Now, given that \(B\) and \(C\) are in the ratio of 5:7, we can write:

\[ B = 5x \]
\[ C = 7x \]

Since the sum of the angles in the triangle is \(180^\circ\), we have:

\[ 120^\circ + 5x + 7x = 180^\circ \]
\[ 12x = 60^\circ \]
\[ x = 5^\circ \]

Now, we can find the values of \(B\) and \(C\):

\[ B = 5x = 5 \times 5^\circ = 25^\circ \]
\[ C = 7x = 7 \times 5^\circ = 35^\circ \]

So, the second largest angle of the triangle is \(35^\circ\).