Question

if $\angle adc$ measures $61^\circ$, what does $\angle abc$ measure?

Explanation:

Step1: Recall the inscribed - angle theorem

The measure of an inscribed angle is half the measure of the central angle that subtends the same arc. Here, \(\angle ADC\) is an inscribed angle and \(\angle ABC\) is a central angle subtending the same arc \(AC\).

Step2: Apply the formula

We know that if \(\theta_{inscribed}\) is the measure of the inscribed angle and \(\theta_{central}\) is the measure of the central angle subtending the same arc, then \(\theta_{inscribed}=\frac{1}{2}\theta_{central}\). Given \(\angle ADC = 61^{\circ}\), and \(\angle ABC\) is the central angle corresponding to the same arc \(AC\) as \(\angle ADC\). So, \(\angle ABC = 2\angle ADC\).

Step3: Calculate the value

Substitute \(\angle ADC=61^{\circ}\) into the formula \(\angle ABC = 2\times61^{\circ}\).

Answer:

\(122\)