Question

in circle o, what is ( moverarc{ae} )?
( 84^circ )
( 96^circ )
( 120^circ )
( 168^circ )

Explanation:

Step1: Recall the secant - secant angle theorem

The measure of an angle formed by two secants intersecting outside a circle is equal to half the difference of the measures of the intercepted arcs. The formula is \(m\angle C=\frac{1}{2}(m\widehat{AE}-m\widehat{BD})\), where \(\angle C\) is the angle formed outside the circle, \(m\widehat{AE}\) is the measure of the larger intercepted arc, and \(m\widehat{BD}\) is the measure of the smaller intercepted arc.

We know that \(m\angle C = 36^{\circ}\) and \(m\widehat{BD}=48^{\circ}\). Let \(x = m\widehat{AE}\).

Step2: Substitute the known values into the formula

Substitute \(m\angle C = 36^{\circ}\), \(m\widehat{BD}=48^{\circ}\) and \(x = m\widehat{AE}\) into the formula \(m\angle C=\frac{1}{2}(m\widehat{AE}-m\widehat{BD})\):

\(36^{\circ}=\frac{1}{2}(x - 48^{\circ})\)

Step3: Solve for \(x\)

First, multiply both sides of the equation by 2 to get rid of the fraction:

\(2\times36^{\circ}=x - 48^{\circ}\)

\(72^{\circ}=x - 48^{\circ}\)

Then, add \(48^{\circ}\) to both sides of the equation:

\(x=72^{\circ}+ 48^{\circ}\)

\(x = 120^{\circ}\)

Answer:

\(120^{\circ}\) (corresponding to the option "120°")