Question

a circle is centered on point b. points a, c and d lie on its circumference. if ∠adc measures 20°, what does ∠abc measure? blank°

Explanation:

Step1: Recall the inscribed angle theorem

The inscribed angle theorem states that an angle subtended by an arc at the center of a circle (central angle) is twice the angle subtended at the circumference (inscribed angle) by the same arc.
Here, \(\angle ADC\) is an inscribed angle subtended by arc \(AC\), and \(\angle ABC\) is the central angle subtended by the same arc \(AC\).

Step2: Apply the theorem to find \(\angle ABC\)

Given \(\angle ADC = 20^\circ\), by the inscribed angle theorem, the central angle \(\angle ABC\) (subtended by arc \(AC\)) is twice the inscribed angle \(\angle ADC\) (subtended by the same arc \(AC\)).
So, \(\angle ABC = 2\times\angle ADC\).
Substitute \(\angle ADC = 20^\circ\) into the formula: \(\angle ABC = 2\times20^\circ = 40^\circ\).

Answer:

\(40\)