Angle 
\[A\] is circumscribed about circle 
\[O\].
What is the measure of 
\[\angle D\]?
 
\[^\circ\]
A circle centered around point O. Point A is outside of the circle. Segment O B and segment O C are both radii of the circle. There are line segments that connect point A to points B and C, creating segment A B and segment A C. Angle B A C measures eighty degrees. There is a point D on the circle that lies opposite to point A. There are chords connecting points B and C to point D, creating segment B D and segment C D.


To solve for the measure of \(\angle D\), we need to apply some principles from circle geometry, particularly the **inscribed angle theorem** and properties of cyclic quadrilaterals.

Here is a breakdown:

1. **Inscribed Angle Theorem**: This theorem states that an inscribed angle is half the measure of the central angle subtended by the same arc. In this case, \(\angle B A C = 80^\circ\). Since \(A\) is an external point and \(B\), \(C\), and \(D\) lie on the circle, this situation forms a **cyclic quadrilateral**.

2. **Cyclic Quadrilateral Property**: In a cyclic quadrilateral, the opposite angles are supplementary. So, \(\angle B A C\) and \(\angle B D C\) are opposite angles. Therefore, the sum of \(\angle B A C\) and \(\angle B D C\) must equal \(180^\circ\).

   \[
   \angle B A C + \angle B D C = 180^\circ
   \]

3. Since \(\angle B A C = 80^\circ\), we can substitute this value into the equation:

   \[
   80^\circ + \angle B D C = 180^\circ
   \]

4. Solving for \(\angle B D C\):

   \[
   \angle B D C = 180^\circ - 80^\circ = 100^\circ
   \]

Thus, the measure of \(\angle D\), which is \(\angle B D C\), is **100°**.

---

Would you like further clarification or details? Here are some related questions for deeper understanding:

1. What is the inscribed angle theorem and how does it apply to other circle problems?
2. Can you explain how cyclic quadrilaterals are identified and their properties?
3. How would the problem change if \(\angle B A C\) were different?
4. What other angles could be calculated in this configuration using circle theorems?
5. How does the location of point \(D\) affect the cyclic quadrilateral's properties?

**Tip**: In any problem involving circles, always consider using the inscribed angle theorem and properties of cyclic quadrilaterals when you see points connected by chords and arcs.

Angle A is circumscribed about circle O. What is the measure of angle D? A circle centered around point O. Point A is outside of the circle. Segment OB and segment OC are both radii of the circle. There are line segments that connect point A to points B and C, creating segment AB and segment AC. Angle BAC measures eighty degrees. There is a point D on the circle that lies opposite to point A. There are chords connecting points B and C to point D, creating segment BD and segment CD.

Explore how to calculate the measure of angle D in a cyclic quadrilateral using circle geometry and the inscribed angle theorem. This problem applies key principles such as the cyclic quadrilateral property and angle sum formula. Ideal for Grades 9-11 students.

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