We are given a diagram with parallel lines cut by a transversal. We need to find the values of $x$, $y$, and $z$.

1. Problem Description

We are given a diagram with parallel lines cut by a transversal. We need to find the values of $x$, $y$, and $z$.

2. Solution Steps

First, we can establish a relationship between $(3y-11)$ and $(y+19)$ because the lines are parallel and these angles are alternate interior angles, which must be equal.

$3y - 11 = y + 19$

Solve for $y$:

$3y - y = 19 + 11$

$2y = 30$

$y = 15$

Next, we observe the right angle in the diagram. The angle $(4z+2)$ and the right angle (90 degrees) form a linear pair. This means they are supplementary, and their sum is 180 degrees. But it looks like it is complementary with $(y+19)$.

Therefore, $(4z+2) + (y+19) = 90$

Substitute $y = 15$:

$4z + 2 + 15 + 19 = 90$

$4z + 36 = 90$

$4z = 90 - 36$

$4z = 54$

$z = \frac{54}{4}$

$z = \frac{27}{2} = 13.5$

Now, let's find the value of the angle $(3y-11)$.

$3y - 11 = 3(15) - 11 = 45 - 11 = 34$

So the angle $(3y-11) = 34$ degrees.

Also, we can find the angle $(y+19)$.

$y+19 = 15+19 = 34$

The angle $(y+19) = 34$ degrees.

The angles $x$ and $(3y-11)$ are supplementary angles. Therefore, $x + (3y-11) = 180$.

Substituting the value of $(3y-11)$:

$x + 34 = 180$

$x = 180 - 34$

$x = 146$

3. Final Answer

$x = 146$

$y = 15$

$z = 13.5$