Question

in the circle below, find the measure of ∠CPE if the measure of arc BE = 100°, the measure of arc BD = 34°, and the measure of arc DC = 96°. The figure is not drawn to scale.

Answer 1

The central angle of a circle is twice any inscribed angle subtended by the same arc.

the angle DPB = the angle CPE

due to the rules of crossing lines.

and either of them is then

(arc angle BD + arc angle CE)/2

we know, that all arc angles in a circle together must be 360°, of course.

so, arc angle CE = 360 - BE - BD - DC =

= 360 - 100 - 34 - 96 = 130°

therefore, we have

CPE = (34 + 130)/2 = 164/2 = 82°