In the diagram, find the value of angle y and angle z.

The angle $y$ is $\angle BCD$. In a cyclic quadrilateral, opposite angles sum to $180^\circ$.
We are given $\angle DAB = 48^\circ$.

We are given $\angle ABC = 46^\circ$. In a cyclic quadrilateral, opposite angles sum to $180^\circ$.

The angle $2x$ is the central angle $\angle AOD$, subtending arc AD.
The angle at the circumference subtending the same arc AD is $\angle ABD$ or $\angle ACD$.
Therefore, $\angle ABD = \frac{1}{2}(2x) = x$ and $\angle ACD = \frac{1}{2}(2x) = x$.

The angle $z$ is formed by the tangent at D and the chord CD.
By the Alternate Segment Theorem, the angle between a tangent and a chord through the point of contact is equal to the angle in the alternate segment.
The angle in the alternate segment subtended by chord CD is $\angle CAD$.
Therefore, $z = \angle CAD$.

Consider $\triangle ACD$. The sum of angles in a triangle is $180^\circ$.
We know $\angle ACD = x$ (from Step 3) and $\angle ADC = 134^\circ$ (from Step 2).

Since $z = \angle CAD$ (from Step 4), we have:

Now, consider $\angle DAB = 48^\circ$.

Angles subtended by the same arc are equal. Arc BC subtends $\angle BAC$ and $\angle BDC$.
So, $\angle BDC = \angle BAC = 2^\circ + x$.
Now, consider $\angle ADC = 134^\circ$.

Angles subtended by the same arc are equal. Arc AB subtends $\angle ADB$ and $\angle ACB$.
So, $\angle ACB = \angle ADB = 132^\circ - x$.
Now, consider $\angle ABC = 46^\circ$.

We know $\angle ABD = x$ (from Step 3).

Angles subtended by the same arc are equal. Arc CD subtends $\angle CAD$ and $\angle CBD$.
So, $\angle CAD = \angle CBD$. This is consistent with $z = \angle CAD$ and $\angle DBC = \angle CBD = 46^\circ - x$.
Thus, $z = 46^\circ - x$.

The problem as stated leads to an identity, meaning $x$ and $z$ cannot be uniquely determined with the given information. This often indicates a missing piece of information or a misinterpretation of the diagram. However, if we assume that the diagram implies that $x$ and $z$ are equal, which is a common simplification in some problems when they are related by the same arc or theorem, then:
If $x = z$, then substituting into $z = 46^\circ - x$:

And since $z = x$, then $z = 23^\circ$.

Given the structure of these problems, it's highly probable that $x$ and $z$ are intended to be equal, or there's a specific relationship that makes them equal. For instance, if chord AD was parallel to chord BC, or if some other specific angle was given. Without additional information, $x$ and $z$ cannot be uniquely determined from the given angles $48^\circ$ and $46^\circ$ alone. However, if we assume the most common interpretation where $x$ and $z$ are related by the alternate segment theorem and central angle theorem, and the problem expects a numerical answer, the only way to resolve the ambiguity is if $x=z$.

Assuming $x=z$:

The values are:
$x = 23^\circ, y = 132^\circ, z = 23^\circ$