We have been given an image of a circle A, where BC || DE , mBC=58° and mDE=142°. We are asked to find the measure of angle CFE.
Since segment DC is parallel to segment DE, so measure of arc BD will be equal to arc CE.
[tex]\widehat{BD}=\widehat{CE}[/tex]
We know that sum of all arcs of a circle is equal to 360 degrees. So we can set an equation as:
[tex]\widehat{BD}+\widehat{CE}+\widehat{BC}+\widehat{DE}=360^{\circ}[/tex]
Since [tex]\widehat{BD}=\widehat{CE}[/tex], so we will get:
[tex]\widehat{CE}+\widehat{CE}+\widehat{BC}+\widehat{DE}=360^{\circ}[/tex]
[tex]2\widehat{CE}+\widehat{BC}+\widehat{DE}=360^{\circ}[/tex]
[tex]2\widehat{CE}+58^{\circ}+142^{\circ}=360^{\circ}[/tex]
[tex]2\widehat{CE}+200^{\circ}=360^{\circ}[/tex]
[tex]2\widehat{CE}+200^{\circ}-200^{\circ}=360^{\circ}-200^{\circ}[/tex]
[tex]2\widehat{CE}=160^{\circ}[/tex]
[tex]\frac{2\widehat{CE}}{2}=\frac{160^{\circ}}{2}[/tex]
[tex]\widehat{CE}=80^{\circ}[/tex]
Therefore, the measure of arc CE is 80 degrees.
We can see that angle CFE is inscribed angle of arc CE, so measure of angle CFE will be half the measure of arc CE.
[tex]\angle CFE=\frac{80^{\circ}}{2}[/tex]
[tex]\angle CFE=40^{\circ}[/tex]
Therefore, the measure of angle CFE is 40 degrees.
