We are given a circle with an external point from which two tangent lines are drawn. We are given that one arc of the circle formed by the points of tangency is $226^\circ$. We need to find the measures of the angles $x$ and $y$ as shown in the diagram.

1. Problem Description

We are given a circle with an external point from which two tangent lines are drawn. We are given that one arc of the circle formed by the points of tangency is $226^\circ$. We need to find the measures of the angles $x$ and $y$ as shown in the diagram.

2. Solution Steps

First, we need to find the measure of the other arc in the circle. Since the total degrees in a circle is $360^\circ$, the measure of the minor arc is $360^\circ - 226^\circ = 134^\circ$.

$x$ is the angle formed by two tangent lines outside the circle. The measure of this angle is half the difference of the intercepted arcs. So we have:

$x = \frac{226^\circ - 134^\circ}{2} = \frac{92^\circ}{2} = 46^\circ$

Now, we need to find the angle $y$. Angle $y$ is the inscribed angle corresponding to the arc which measures $134^\circ$. The measure of an inscribed angle is half the measure of its intercepted arc. So we have:

$y = \frac{134^\circ}{2} = 67^\circ$

3. Final Answer

$x = 46^\circ$

$y = 67^\circ$