Note that the semicircle JEF and the minor arc FG are adjacent arcs. By the arc addition postulate, we know that the measure of the arc JEG is the sum of the measures of these two adjacent arcs.
mJEG=m JEF+m FGIn the above equation, we are missing the measure of FG. Therefore, we first need to find the measure of FG. Looking at the diagram, we note that the central angles ∠FOG and ∠GOH are congruent angles. Therefore they have the same measure. Let x be the measure of these two angles.
Moreover, we know that a central angle and its intercepted arc have the same measure. Therefore, since FG and GH are the intercepted arcs of ∠FOG and ∠GOH, respectively, we know that mFG= x and mGH= x.
Note that FG and GH are adjacent arcs. Therefore, by the Arc Addition Postulate, we can say that the measure of FH is the sum of the measures of these two arcs.
mFH=FG+mGH
⇓
mFH= x+ x
mFH= 2x
Let's add this information to our diagram.
Also note, that FH and HJ are two adjacent arcs that form a semicircle. Moreover, the measure of a semicircle is 180^(∘). Therefore, by the Arc Addition Postulate, we know that the sum of mFH and mHJ equals 180^(∘).
mFH+mHJ=180^(∘)
Let's substitute mFH= 2x and mHJ= 70 in the above equation, and solve for x.
