The problem asks to find the value of $x$ in each of the three figures, given that the segments that appear to be tangent are indeed tangent.

1. Problem Description

The problem asks to find the value of $x$ in each of the three figures, given that the segments that appear to be tangent are indeed tangent.

2. Solution Steps

Figure 1:

The line is tangent to the circle at point A. Therefore, the radius OA is perpendicular to the tangent line at A. This means the angle OAB is a right angle, so it measures $90^\circ$. The triangle OAB is a right triangle.

The angle at O is $40^\circ$ as shown in the figure. The sum of angles in triangle OAB is $180^\circ$.

So, $40^\circ + 90^\circ + x = 180^\circ$.

Therefore $x = 180^\circ - 90^\circ - 40^\circ = 50^\circ$.

Figure 2:

The two tangent segments from the same external point to a circle are congruent.

Also, the two segments from the center to the points of tangency are radii, so they are congruent. The two segments form a quadrilateral.

The angles formed by radii and the tangents are right angles, each equal to $90^\circ$. The tangents form an angle of $33^\circ$.

The sum of the interior angles of the quadrilateral is $360^\circ$.

So $90^\circ + 90^\circ + 33^\circ + x = 360^\circ$.

Therefore $x = 360^\circ - 90^\circ - 90^\circ - 33^\circ = 147^\circ$.

Figure 3:

Two tangent segments are drawn from the same external point to a circle. The two segments from the same external point to the circle are congruent. So the triangle formed by the two tangent segments and the line segment connecting the tangent points on the circle is an isosceles triangle. Let the angle at the center be $x^\circ$. Since the lines connecting the center to the tangent points are perpendicular to the tangent lines, and the angle formed by the tangent lines is $40^\circ$, we have two right angles and an angle of $40^\circ$ in a quadrilateral. Let the two angles in the isosceles triangle at the base be $y^\circ$.

Then, we have $2y^\circ + 40^\circ = 180^\circ$. Thus, $2y^\circ = 140^\circ$, and $y^\circ = 70^\circ$.

The lines connecting the center of the circle to the tangent point bisect the angle between the radii and the point where the tangent lines meet. The angle between the two radii is $x^\circ$.

Consider the quadrilateral. We have two right angles at the intersection of the radii and tangent segments, so the sum of the central angle $x^\circ$ and the $40^\circ$ tangent angle is $180^\circ$.

Thus, $x^\circ + 40^\circ = 180^\circ$, which is incorrect.

The angle between the tangents is 40 degrees. Draw the segment from the center to the point where the tangent lines meet. This bisects the 40 degree angle, making it 20 degrees. Draw the radius to each tangent point. This is a right angle. Then consider the triangle formed by one of the tangent lines, the segment from the center to the tangent point, and the line from the center to where the tangent lines meet. This is a right triangle. The angle at the center is 90-20 = 70 degrees. The full angle at the center, $x$ is twice this, so $x = 2 * 70 = 140$.

3. Final Answer

Figure 1: $x = 50$

Figure 2: $x = 147$

Figure 3: $x = 140$