Question

question 5 (10 points) given ( moverarc{eb} = 128^circ ) and ( mangle efb = 46^circ ), determine the measure of the arc ( moverarc{cd} ). (you may assume that point a is the center of the circle.) (figure may not be drawn to scale.)
a)
b)
c)
d)

Explanation:

Step1: Recall the theorem for angle formed by two secants

The measure of an angle formed by two secants outside a circle is half the difference of the measures of the intercepted arcs. So, \( m\angle DFE=\frac{1}{2}(m\widehat{EB}-m\widehat{CD}) \).

Step2: Substitute the known values

We know \( m\widehat{EB} = 128^\circ \) and \( m\angle DFE=46^\circ \). Substitute these into the formula: \( 46^\circ=\frac{1}{2}(128^\circ - m\widehat{CD}) \).

Step3: Solve for \( m\widehat{CD} \)

First, multiply both sides by 2: \( 46^\circ\times2=128^\circ - m\widehat{CD} \), which gives \( 92^\circ = 128^\circ - m\widehat{CD} \). Then, rearrange the equation: \( m\widehat{CD}=128^\circ - 92^\circ \). Calculate the right - hand side: \( m\widehat{CD}=36^\circ \).

Answer:

\( 36^\circ \)