Question

as shown in the diagram, \\( \overline { d a } \\) and \\( \overline { d c } \\) are tangent to circle \\( k \\). points \\( a, b \\), and \\( c \\) lie on circle \\( k \\).

consider the relationships between the central angle, \\( \theta \\), of the circle and the angles and arcs shown in the diagram.
move phrases to the blanks to complete the sentences.

the central angle, \\( \theta \\), is ______ minor arc \\( \widehat { a c } \\).
the central angle, \\( \theta \\), is ______ \\( \angle a b c \\).
the central angle, \\( \theta \\), is ______ \\( \angle a d c \\).

Explanation:

Step1: Central angle and arc relationship

By the definition of a central angle, the central angle \(\theta\) is equal in measure to the minor arc \(\widehat{AC}\).

Step2: Central angle and inscribed angle relationship

The measure of an inscribed angle (\(\angle ABC\)) is half the measure of the central angle subtended by the same arc. So the central angle \(\theta\) is twice \(\angle ABC\).

Step3: Tangent - radius and angle sum in quadrilateral

Since \(DA\) and \(DC\) are tangents, \(\angle DAK=\angle DCK = 90^{\circ}\) (tangent - radius property). In quadrilateral \(DAKC\), \(\angle D+\angle A KC=180^{\circ}\) (sum of interior angles of a quadrilateral \(= 360^{\circ}\), and \(\angle DAK=\angle DCK = 90^{\circ}\)). So the central angle \(\theta\) is supplementary to \(\angle ADC\).

Answer:

The central angle, \(\theta\), is equal in measure to minor arc \(\widehat{AC}\).
The central angle, \(\theta\), is twice \(\angle ABC\).
The central angle, \(\theta\), is supplementary to \(\angle ADC\).