In the given diagram, $WY$ and $WZ$ are straight lines, and $O$ is the center of the circle on the left. The angle $OWX$ is $48^\circ$. We are asked to find the angle $XYZ$.

1. Problem Description

In the given diagram, $WY$ and $WZ$ are straight lines, and $O$ is the center of the circle on the left. The angle $OWX$ is $48^\circ$. We are asked to find the angle $XYZ$.

2. Solution Steps

Since $O$ is the center of the circle on the left and $WOMZ$ is a straight line, the points $W$, $O$, $M$, and $Z$ are collinear.

In triangle $OWX$, since $OW$ and $OX$ are radii of the same circle, $OW = OX$. Therefore, triangle $OWX$ is an isosceles triangle.

Since $OWX$ is an isosceles triangle with $OW = OX$, the angles opposite these sides are equal, i.e., $\angle OWX = \angle OXW = 48^\circ$.

The sum of angles in a triangle is $180^\circ$. In triangle $OWX$, we have

$\angle WOW + \angle OXW + \angle XOW = 180^\circ$

$48^\circ + 48^\circ + \angle XOW = 180^\circ$

$96^\circ + \angle XOW = 180^\circ$

$\angle XOW = 180^\circ - 96^\circ$

$\angle XOW = 84^\circ$

Since $\angle XOW$ and $\angle XOM$ are supplementary angles, i.e., they form a straight line,

$\angle XOW + \angle XOM = 180^\circ$

$84^\circ + \angle XOM = 180^\circ$

$\angle XOM = 180^\circ - 84^\circ$

$\angle XOM = 96^\circ$

Since $XMYZ$ are points on the circle on the right, $XMZY$ is a cyclic quadrilateral. The exterior angle of a cyclic quadrilateral is equal to the interior opposite angle. Also, the sum of opposite angles in a cyclic quadrilateral is $180^\circ$. The angle at the center of a circle is twice the angle at the circumference when subtended by the same arc.

Consider the circle passing through $X$, $Y$, $Z$. Since $XMZ$ is a straight line, $XZ$ is a chord of the circle.

$\angle XOM$ is the angle at the center subtended by the chord $XZ$.

$\angle XYZ$ is the angle at the circumference subtended by the chord $XZ$.

$\angle XOM = 2\angle XYZ$

$96^\circ = 2 \angle XYZ$

$\angle XYZ = \frac{96^\circ}{2}$

$\angle XYZ = 48^\circ$

3. Final Answer

$48^\circ$