Question

Find the measure of angle CEF, angle D, arc AC, and angle EBD.

Answer 1

For the all questions we use the theorem of chords and arcs and get that for the first question:

`\measuredangle CEF=\text{ }(1)/(2)(30^(\circ)+58^(\circ))=44^(\circ)`

Now, to find angle D:

`\measuredangle D=(1)/(2)(arcAB\text{ -arcGF)=}(1)/(2)(58^(\circ)-20^(\circ))=19^(\circ)`

Next for arcAC we use that WC and WA are tangent to the circle

O is the center of the circle ( O is not E). Now we recall that the inner angles of a quadrilateral is 360 so

`\measuredangle\alpha=360-80-90-90=100`

By the definition of arcAC

`arc\text{AC}=\measuredangle\alpha=100`

Finally we use the fact that the value of inscribed angles is half of the value of the central angle, using this we get :

`\measuredangle EBD=\measuredangle CBG=(1)/(2)\measuredangle COG\text{ = }(1)/(2)(\text{arcFC}+\text{arcGF)}=(1)/(2)50^(\circ)=25^(\circ)`