**Header:** Tangent Angle and Circle Geometry Problem
**Solution:**
Since PA and PB are tangents to the circle with center O, we know:
- ∠OAP = ∠OBP = 90° (radius perpendicular to tangent)
- OA = OB (radii of the same circle)
- PA = PB (tangents from same external point)
In quadrilateral OAPB:
∠APB + ∠OAP + ∠OBP + ∠AOB = 360°
50° + 90° + 90° + ∠AOB = 360°
∠AOB = 130°
In triangle OAB:
Since OA = OB, triangle OAB is isosceles
∠OAB = ∠OBA
∠OAB + ∠OBA + ∠AOB = 180°
2∠OAB + 130° = 180°
2∠OAB = 50°
∠OAB = 25°
**Answer:** 25 degrees
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We have a circle with center O. PA and PB are tangents to the circle from external point P. The angle APB is given as 50 degrees. We need to find the measure of angle OAB.
Let's examine the key properties of tangents. First, a tangent is always perpendicular to the radius at the point of contact. Second, two tangents drawn from an external point to a circle are equal in length. Third, there's a relationship between the angle formed by the tangents and the central angle. Since angle APB is 50 degrees, the central angle AOB is 130 degrees, because they are supplementary in this configuration.
Now let's analyze quadrilateral PAOB. We know that angle OAP equals 90 degrees because the tangent is perpendicular to the radius. Similarly, angle OBP equals 90 degrees. We're given that angle APB equals 50 degrees. Since the sum of angles in any quadrilateral is 360 degrees, we can find angle AOB. It equals 360 minus 90 minus 90 minus 50, which gives us 130 degrees.
Now we can find angle OAB by focusing on triangle OAB. Since OA and OB are both radii of the circle, they are equal in length, making triangle OAB isosceles. In an isosceles triangle, the base angles are equal, so angle OAB equals angle OBA. We know angle AOB is 130 degrees. Using the fact that angles in a triangle sum to 180 degrees, we get 130 plus 2 times angle OAB equals 180. Solving this equation: 2 times angle OAB equals 50, so angle OAB equals 25 degrees.
Let's summarize our complete solution. We were given that angle APB equals 50 degrees. First, we found that the central angle AOB equals 130 degrees using the properties of the quadrilateral PAOB. Then, we recognized that triangle OAB is isosceles since OA and OB are both radii. Finally, we used the angle sum property of triangles to find that angle OAB equals 25 degrees. This is our final answer.