Question

find the ( mangle aed ).

Explanation:

Step1: Recall the theorem for intersecting chords

When two chords intersect in a circle, the measure of the angle formed is equal to half the sum of the measures of the intercepted arcs. So, \( m\angle AED=\frac{1}{2}(m\overset{\frown}{AD} + m\overset{\frown}{BC}) \). But first, we need to find the measure of the arc opposite to the given arcs. The total circumference of a circle is \( 360^\circ \), so the measure of arc \( AB \) and arc \( DC \) can be found, but actually, we know two arcs: \( 92^\circ \) (arc \( AD \)) and we need to find the other arc. Wait, the given arcs are \( 92^\circ \) (arc \( AD \)) and \( 140^\circ \) (arc \( BC \))? Wait, no, when two chords intersect at \( E \), the arcs intercepted by \( \angle AED \) are arc \( AD \) and arc \( BC \). Wait, actually, the formula is \( m\angle AED=\frac{1}{2}(m\overset{\frown}{AD} + m\overset{\frown}{BC}) \). Wait, but first, let's check the total of the arcs. The sum of all arcs in a circle is \( 360^\circ \). Wait, the two given arcs: one is \( 92^\circ \) (arc \( AD \)) and another is \( 140^\circ \) (arc \( BC \))? Wait, no, maybe I got the arcs wrong. Wait, the chords are \( AB \) and \( CD \) intersecting at \( E \). So the arcs intercepted by \( \angle AED \) are arc \( AD \) and arc \( BC \). Wait, the measure of \( \angle AED \) is half the sum of the measures of the arcs intercepted by it and its vertical opposite angle. So, first, let's confirm the arcs. The arc \( AD \) is \( 92^\circ \), and the arc \( BC \) is \( 140^\circ \)? Wait, no, maybe the other arcs: the sum of arc \( AD \) and arc \( BC \) and the other two arcs (arc \( AB \) and arc \( DC \)) should be \( 360^\circ \). But actually, the formula for the angle formed by two intersecting chords is \( m\angle AED=\frac{1}{2}(m\overset{\frown}{AD} + m\overset{\frown}{BC}) \). Wait, let's plug in the values. Wait, arc \( AD = 92^\circ \), arc \( BC = 140^\circ \)? Wait, no, that can't be, because \( 92 + 140 = 232 \), half of that is \( 116 \), which is one of the options. Wait, let's check:

Wait, the formula is \( m\angle AED=\frac{1}{2}(m\overset{\frown}{AD} + m\overset{\frown}{BC}) \). Given \( m\overset{\frown}{AD} = 92^\circ \) and \( m\overset{\frown}{BC}=140^\circ \). Then, \( m\angle AED=\frac{1}{2}(92^\circ + 140^\circ) \).

Step2: Calculate the sum of the arcs

First, sum the measures of the arcs: \( 92^\circ+ 140^\circ = 232^\circ \).

Step3: Apply the formula for the angle

Then, \( m\angle AED=\frac{1}{2}\times232^\circ = 116^\circ \).

Answer:

\( 116^\circ \) (corresponding to the option "116°")