Angle
\[A\] is circumscribed about circle
\[O\].
What is the measure of
\[\angle A\]?

\[^\circ\]
A circle centered around point O. Segment C O and segment C B are both radii. There are tangent lines from points C and B on the circle, making segment C A and segment A B. The four lines create quadrilateral C O B A, where angle C O B is ninety-two degrees.

To find the measure of angle \( \angle A \), let's use some properties of the circle and the angles formed by tangent lines and radii.

1. **Tangents and Radii**: The tangent to a circle at any point is perpendicular to the radius at that point. Thus, \(\angle CAO\) and \(\angle ABO\) are right angles (90° each).

2. **Quadrilateral Angle Sum**: The sum of the interior angles in any quadrilateral is \(360^\circ\).

Given:
- \( \angle COB = 92^\circ \) (Central angle formed by radii \(OC\) and \(OB\))
- \(\angle CAO = 90^\circ\) (Tangent at \(C\) and radius \(OC\))
- \(\angle ABO = 90^\circ\) (Tangent at \(B\) and radius \(OB\))

To find \( \angle A \), we calculate the sum of the other three angles and subtract from \(360^\circ\):

\[
\angle A = 360^\circ - (\angle CAO + \angle ABO + \angle COB)
\]

Substituting the given angles:

\[
\angle A = 360^\circ - (90^\circ + 90^\circ + 92^\circ)
\]

\[
\angle A = 360^\circ - 272^\circ
\]

\[
\angle A = 88^\circ
\]

Therefore, the measure of \( \angle A \) is \( 88^\circ \).

Would you like more details or have any other questions?

Here are 8 relative questions to further explore this topic:
1. What is the significance of a tangent being perpendicular to the radius at the point of contact?
2. How can the inscribed angle theorem be used in problems involving circles?
3. What is the formula for finding the angles of a quadrilateral?
4. How would the measure of \( \angle A \) change if \( \angle COB \) were increased or decreased?
5. How do you find the arc length subtended by an angle in a circle?
6. Can the properties of a circle be applied to find angles in other polygons?
7. What are the steps to constructing a tangent line from a point to a circle?
8. How does the radius of the circle affect the calculations in this problem?

**Tip:** Remember, the sum of the interior angles in any polygon can be determined using the formula \((n-2) \times 180^\circ\), where \(n\) is the number of sides in the polygon.

Discover how to calculate the measure of angle A in a circumscribed circle problem. This solution involves understanding circle geometry, angles formed by tangents and radii, and properties of quadrilaterals. Suitable for advanced high school students.

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