Pearson Geometry Common Core, 2011
PG
Pearson Geometry Common Core, 2011 View details
6. Circles and Arcs
Continue to next subchapter

Exercise 41 Page 655

The Arc Addition Postulate tells us that the measure of the arc formed by two adjacent arcs is the sum of the measures of the two arcs.

235

Practice makes perfect

In the given diagram, we want to find mJEG.

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 FG

In 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.
mFH+mHJ=180
2x+ 70=180
2x=110
x=55
We found that x= 55. Therefore, the arc measure of FG is 55^(∘). Let's substitute this value in the above equation, and solve for mJEG.
mJEG=mJEF+mFG
mJEG= 180+ 55
mJEG=235
We found that JEG measures 235^(∘).