Find the value of angle e.

Let's solve for angle 'e'. **Understanding the properties:** 1. A tangent to a circle is perpendicular to the radius at the point of tangency. 2. The sum of angles in a quadrilateral is 360 degrees. **Solution:** 1. **Identify the quadrilateral:** In the given diagram, we can identify a quadrilateral formed by the two tangent lines and the two radii that meet the tangent lines at the points of tangency. Let's call the center of the circle O, and the points where the tangents touch the circle A and B. The point where the tangents meet outside the circle is where angle 'e' is. Let's call that point C. 2. **Angles at the point of tangency:** Since the radii OA and OB are perpendicular to the tangents at points A and B, \(\angle OAC = 90^\circ\) and \(\angle OBC = 90^\circ\). 3. **Central Angle:** The angle at the center of the circle is given as \(138^\circ\). Therefore, \(\angle AOB = 138^\circ\). 4. **Angles in Quadrilateral:** In quadrilateral OACB, the sum of all angles is \(360^\circ\). Therefore, \[\angle OAC + \angle OBC + \angle AOB + \angle ACB = 360^\circ\] Substituting the known values: \[90^\circ + 90^\circ + 138^\circ + e = 360^\circ\] \[318^\circ + e = 360^\circ\] \[e = 360^\circ - 318^\circ\] \[e = 42^\circ\] Therefore, the value of angle 'e' is **42 degrees**.