Question

using circle c, find the measure of each arc or angle named. next to each measure, write the name of the arc or angle. ed is a diameter.

m∠fcd = 60°
central angle
mae = 45°

madf = 135 + 60 = 195

mad = 135

m∠acd = 195

m∠acf = 120

mafd =

mfde =

Explanation:

Step1: Identify the measure of arc \(AE\)

The measure of arc \(AE\) is given as \(45^{\circ}\) (central angle theorem: the measure of an arc is equal to the measure of its central angle).

Step2: Calculate \(m\overset{\frown}{ADF}\)

\(m\overset{\frown}{ADF}=m\overset{\frown}{AD}+m\overset{\frown}{DF}\). Since \(m\overset{\frown}{AD} = 135^{\circ}\) (central angle \( \angle ACD=135^{\circ}\)) and \(m\overset{\frown}{DF}=60^{\circ}\) (central angle \( \angle FCD = 60^{\circ}\)), then \(m\overset{\frown}{ADF}=135^{\circ}+60^{\circ}=195^{\circ}\)

Step3: Calculate \(m\overset{\frown}{AD}\)

The central angle \( \angle ACD = 135^{\circ}\), so by the central angle theorem \(m\overset{\frown}{AD}=135^{\circ}\)

Step4: Calculate \(m\angle ACD\)

Given from the diagram, \(m\angle ACD = 135^{\circ}\) (central angle)

Step5: Calculate \(m\angle ACF\)

\(m\angle ACF=m\angle ACD + m\angle FCD\). Substituting \(m\angle ACD = 135^{\circ}\) and \(m\angle FCD=60^{\circ}\), we get \(m\angle ACF=135^{\circ}+60^{\circ}=195^{\circ}\)

Step6: Calculate \(m\overset{\frown}{AFD}\)

\(m\overset{\frown}{AFD}=m\overset{\frown}{AF}+m\overset{\frown}{FD}\). \(m\overset{\frown}{AF}=360^{\circ}-(45^{\circ}+120^{\circ}+60^{\circ}+135^{\circ})= 360^{\circ}-360^{\circ}+m\overset{\frown}{AF}\). Another way: \(m\overset{\frown}{AFD}=m\angle ACF\) (central angle - arc relationship). Since \(m\angle ACF = 195^{\circ}\), \(m\overset{\frown}{AFD}=195^{\circ}\)

Step7: Calculate \(m\overset{\frown}{FDE}\)

\(m\overset{\frown}{FDE}=m\overset{\frown}{FD}+m\overset{\frown}{DE}\). \(m\overset{\frown}{FD} = 60^{\circ}\), \(m\overset{\frown}{DE}=180^{\circ}-45^{\circ}=135^{\circ}\). So \(m\overset{\frown}{FDE}=60^{\circ}+135^{\circ}=195^{\circ}\)

Answer:

\(m\overset{\frown}{AE} = 45^{\circ}\), arc \(AE\)
\(m\overset{\frown}{ADF}=195^{\circ}\), arc \(ADF\)
\(m\overset{\frown}{AD}=135^{\circ}\), arc \(AD\)
\(m\angle ACD = 135^{\circ}\), \(\angle ACD\)
\(m\angle ACF=195^{\circ}\), \(\angle ACF\)
\(m\overset{\frown}{AFD}=195^{\circ}\), arc \(AFD\)
\(m\overset{\frown}{FDE}=195^{\circ}\), arc \(FDE\)