Question

what is the measure of arc bec in circle d? 134° 150° 209° 210°

Explanation:

Step1: Find the measure of arc \(AC\)

The measure of an inscribed angle is half the measure of its intercepted arc. Given \(\angle ABC = 75^{\circ}\), the intercepted arc \(AC\) has a measure \(m\overset{\frown}{AC}=2\times\angle ABC\).
So, \(m\overset{\frown}{AC}=2\times75^{\circ} = 150^{\circ}\).

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

The measure of arc \(AB\) is given as \(76^{\circ}\).

Step3: Calculate the measure of arc \(BEC\)

The sum of the measures of the arcs of a circle is \(360^{\circ}\). Let \(m\overset{\frown}{BEC}\) be the measure of arc \(BEC\). Then \(m\overset{\frown}{BEC}=360^{\circ}-(m\overset{\frown}{AB}+m\overset{\frown}{AC})\).
Substitute \(m\overset{\frown}{AB} = 76^{\circ}\) and \(m\overset{\frown}{AC}=150^{\circ}\) into the formula: \(m\overset{\frown}{BEC}=360^{\circ}-(76^{\circ}+150^{\circ})=360^{\circ}-226^{\circ}=134^{\circ}\).

Answer:

\(134^{\circ}\)