Question

segments ac and bd are diameters of circle o.
what is the measure of \\(\overarc{adb}\\)?
\\(146^{\circ}\\)
\\(107^{\circ}\\)
\\(253^{\circ}\\)
\\(287^{\circ}\\)

Explanation:

Step1: Recall circle's total degrees

A circle has \( 360^\circ \). The central angle for arc \( AD \) is \( 73^\circ \) (given \( \angle AOD = 73^\circ \)).

Step2: Find the measure of arc \( ADB \)

Arc \( ADB \) is the major arc from \( A \) to \( B \) through \( D \). First, find the measure of the minor arc \( AB \). Since \( AC \) and \( BD \) are diameters, \( \angle AOB \) is supplementary to \( \angle AOD \)? Wait, no—actually, \( \angle AOD = 73^\circ \), so \( \angle DOB = 180^\circ - 73^\circ = 107^\circ \)? Wait, no, better: The measure of arc \( ADB \) is the sum of arc \( AD \) and arc \( DB \)? Wait, no, arc \( ADB \) is from \( A \) to \( D \) to \( B \)? Wait, no, the notation \( \widehat{ADB} \) means the arc from \( A \) to \( B \) passing through \( D \). So the central angle for arc \( ADB \) would be \( 360^\circ - \) the central angle of the minor arc \( AB \). Wait, first, find the central angle of minor arc \( AB \). Since \( AC \) and \( BD \) are diameters, \( \angle AOD = 73^\circ \), so \( \angle BOC = 73^\circ \) (vertical angles), and \( \angle AOB = 180^\circ - 73^\circ = 107^\circ \)? Wait, no, \( \angle AOD = 73^\circ \), so \( \angle DOB = 180^\circ - 73^\circ = 107^\circ \)? Wait, maybe better: The measure of arc \( ADB \) is \( 360^\circ - \) the measure of arc \( AB \). Wait, no, let's see: The circle is \( 360^\circ \). The arc \( ADB \) is the major arc from \( A \) to \( B \) through \( D \). The minor arc \( AB \) would have a central angle of \( 180^\circ - 73^\circ = 107^\circ \)? Wait, no, \( \angle AOD = 73^\circ \), so \( \angle AOB = 180^\circ - 73^\circ = 107^\circ \)? Wait, no, \( BD \) is a diameter, so \( \angle BOD = 180^\circ \). Wait, \( \angle AOD = 73^\circ \), so \( \angle AOB = \angle BOD - \angle AOD = 180^\circ - 73^\circ = 107^\circ \). Then the major arc \( ADB \) would be \( 360^\circ - 107^\circ = 253^\circ \)? Wait, no, wait: Arc \( ADB \) is from \( A \) to \( D \) to \( B \), so that's arc \( AD \) (73°) plus arc \( DB \) (180°)? Wait, no, \( DB \) is a diameter? Wait, \( BD \) is a diameter, so arc \( BD \) is \( 180^\circ \). Wait, \( A \) to \( D \) is 73°, \( D \) to \( B \) is 180°, so total arc \( ADB \) is \( 73^\circ + 180^\circ = 253^\circ \). Yes, that makes sense. So \( 73^\circ + 180^\circ = 253^\circ \).

Answer:

253°