b The measure of the major arc is the difference of 360^(∘) and the measure of the related minor arc.
C
c Use the Arc Addition Postulate.
D
d The measure of the major arc is the difference of 360^(∘) and the measure of the related minor arc.
E
e Use the Arc Addition Postulate.
F
f The measure of the major arc is the difference of 360^(∘) and the measure of the related minor arc.
A
a 103^(∘)
B
b 257^(∘)
C
c 196^(∘)
D
d 305^(∘)
E
e 79^(∘)
F
f 281^(∘)
Practice makes perfect
a We are asked to determine the measure of arc AE.
Since the arc is described only with the endpoints A and E, it is a minor arc. Let's highlight AE on the given circle graph.
We can see that AE is formed by the arcs AH, HG, GF, and FE. Since the Arc Addition Postulate holds true for any number of adjacent arcs, the measure of AE is equal to the sum of measures of AH, HG, GF, and FE.
mAE=mAH+mHG+mGF+mFE
From the circle graph we know that mAH=17^(∘), mHG=26^(∘), mGF=28^(∘), and mFE=32^(∘). Let's substitute these values into the above expression and simplify.
b We are asked to determine the measure of arc ACE.
Since the arc is described with the endpoints A and E and a point on the arc C, it is a major arc. Let's highlight ACE on the given circle graph.
We can see that the minor arc related to ACE is AE. Recall that the measure of the major arc is the difference of 360^(∘) and the measure of the related minor arc.
mACE=360^(∘)-mAE
From Part A we know that the measure of AE is 103^(∘). Let's find the measure of ACE.
c We are asked to determine the measure of arc GDC.
Since the arc is described with the endpoints G and C and a point on the arc D, it is a major arc. Let's highlight GDC on the given circle graph.
We can see that GDC is formed by the arcs GF, FE, ED, and DC. Since the Arc Addition Postulate holds true for any number of adjacent arcs, the measure of GDC is equal to the sum of measures of GF, FE, ED, and DC.
mGDC = mGF +mFE +mED + mDC
From the circle graph we know that GF=28^(∘), FE=32^(∘), ED=47^(∘), and DC= 89^(∘). Let's substitute these values into the above expression and simplify.
d We are asked to determine the measure of arc BHC.
Since the arc is described with the endpoints B and C and a point on the arc H, it is a major arc. Let's highlight BHC on the given circle graph.
We can see that the minor arc related to BHC is BC. Recall that the measure of a major arc is the difference of 360^(∘) and the measure of the related minor arc.
mBHC=360^(∘)-mBC
From the circle graph we know that mBC=55^(∘). Let's find the measure of BHC.
e We are asked to determine the measure of arc FD.
Since the arc is described only with the endpoints F and D it is a minor arc. Let's highlight FD on the given circle graph.
We can see that FD is formed by the arcs FE and DE. Therefore, the measure of FD is the sum of the measures of FE and DE by the Arc Addition Postulate.
mFD=mFE+mDE
From the circle graph we know that mFE=32^(∘) and mDE=47^(∘). Let's substitute these values into the above expression and simplify.
f We are asked to determine the measure of arc FBD.
Since the arc is described with the endpoints F and D and a point on the arc B, it is a major arc. Let's highlight FBD on the given circle graph.
We can see that the minor arc related to FBD is FD. Recall that the measure of a major arc is the difference of 360^(∘) and the measure of the related minor arc.
mFBD=360^(∘)-mFD
From Part E we know that the measure of FD is 79^(∘). Let's find the measure of FBD.
