In a circle with the center and radius OA, we extend this radius to the point B. From the point B on the desired tangent of the circle, we drop the perpendicular BD. If the angle AOB is 34 degrees, how many degrees is the angle OAD?


To solve this problem, let's break it down step by step using the geometric properties of circles, angles, and tangents.

### Step 1: Understanding the Problem
- \(O\) is the center of the circle.
- \(A\) is a point on the circle, so \(OA\) is the radius.
- \(B\) is a point outside the circle, and \(AB\) is an extended radius.
- A tangent is drawn from point \(B\) to the circle, touching the circle at point \(D\).
- \(BD\) is perpendicular to the radius \(OD\) at the point of tangency.
- \( \angle AOB = 34^\circ \), and we need to find the angle \( \angle OAD \).

### Step 2: Analyzing the Geometry
1. **Circle Center \(O\), Radius \(OA\)**:
   - \(OA = OD\) because both are radii of the circle.
   
2. **Angle Relationships**:
   - Since \(AB\) is a straight line, \( \angle OAB = \angle AOB = 34^\circ \).
   - \(BD\) is a tangent, so \( \angle ODB = 90^\circ \) (because the radius \(OD\) is perpendicular to the tangent at the point of tangency \(D\)).

### Step 3: Angle \( \angle OAD \)
To find \( \angle OAD \), note the following:
- Since \(BD\) is tangent to the circle at \(D\) and \(OD\) is the radius, \( \angle OBD\) is 90°.
- \( \angle OAD \) is the angle subtended by the arc \(AD\) at the point \(O\).

Now, use the fact that the sum of the angles in triangle \(OAB\) (considering \(O\), \(A\), and \(B\) as vertices) is 180°:

\[
\angle OAB + \angle OBA + \angle AOB = 180^\circ
\]
Since \( \angle OAB = \angle OBA \) (due to symmetry in isosceles triangle \(OAB\)), and \( \angle AOB = 34^\circ \),

\[
2 \times \angle OAB + 34^\circ = 180^\circ
\]
\[
2 \times \angle OAB = 146^\circ \quad \Rightarrow \quad \angle OAB = 73^\circ
\]

Now, since \(OA\) is the radius and \(AD\) is a chord, \( \angle OAD \) will be half of \( \angle OAB \) because of the Inscribed Angle Theorem (angle subtended by the same chord at the center and on the circle):

\[
\angle OAD = \frac{1}{2

Explore how to calculate angles in circle geometry with tangents using a step-by-step solution. This problem involves finding the angle OAD given the angle AOB is 34 degrees. Learn about tangent lines, circle properties, and the Inscribed Angle Theorem. Suitable for high school students.

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