Question

question 9 of 10
what is the measure of \\( \angle a b c \\)?
a. \\( 126 ^ { \circ } \\)
b. \\( 63 ^ { \circ } \\)
c. \\( 44 ^ { \circ } \\)
d. \\( 82 ^ { \circ } \\)

Explanation:

Step1: Use the property of vertical angles and the sum of angles around a point

The sum of angles around a point \(B\) in a circle is \(360^{\circ}\). Also, \(\angle ABE\) and \(\angle CBD\) are vertical angles. Let \(\angle ABC = x\). We know that the measure of an arc is equal to the measure of the central angle that subtends it. The arc \(AE\) has a measure such that the central - angle relationships can be used. Another way is to note that \(\angle ABC\) and \(\angle EBD\) are vertical angles. The sum of the arcs \(AC + CE+ED + DA=360^{\circ}\). But more simply, since \(\angle ABC\) and \(\angle EBD\) are vertical angles. The measure of \(\angle EBD\) can be found using the fact that the sum of the arcs \(AC = 82^{\circ}\) and \(DE=44^{\circ}\). The measure of \(\angle ABC\) (a central angle) is related to the non - adjacent arcs. The formula for the measure of an angle formed by two intersecting chords (in this case, since \(AB\) and \(CB\) are chords and \(B\) is the center, it's a central - angle case. But if we consider the general formula for the angle formed by two lines intersecting at the center (a special case of the angle - between - chords formula where the angle is a central angle). The measure of \(\angle ABC=\frac{1}{2}(m\overset{\frown}{AC}+m\overset{\frown}{DE})\).

Step2: Calculate the value of \(\angle ABC\)

Substitute \(m\overset{\frown}{AC} = 82^{\circ}\) and \(m\overset{\frown}{DE}=44^{\circ}\) into the formula \(\angle ABC=\frac{1}{2}(82^{\circ}+44^{\circ})\).

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Answer:

B. \(63^{\circ}\)