Download or mzG:i@A-*70

Angles Associated with a Circle
In I and L deterrnite wIAGB in terms of AB and CD.
@
Uint: Construct BD.
IADB and ICBD are inscribed angles.
wZADB
l^
: L.nAB
2
(The measure of an inscribed angle is
of the meosure of its intercepted arc.)
half
I
and ruIDBC
wZDBC
: !.*6
2
: nZADB +mZG
l^t^
(The measure of the exterior ongle of a
triangle is equal to the sum of the
measures of the two remote interior
angles.)
L.mCD:!-.mAB+mZG
(Substitution)
t^t^
mZG:1.mCD-L.mAB
(Subtraction)
22
Ir
or mzG:i@A-*70)
t]-
Angles Associated with a Circle
@ Hint: Construct ,BC.
nZACB
:
l^
L-mAB
(The measure of an inscribed angle is
of the measure of its intercepted arc.)
2
l^
half
wICBD : L-mCD
2
m/.AGB: wIACB
I
+
n
nIDBC
(The measure of an exterior angle of a
triangle is equal to the sum of the measures
of the remote interior angles.)
I
^
nZ.AGB:L.mAB+L.mCD
(Substitution)
22
or
mzAGB
:
I
|@n * *6)
AG and BG
are tangent
then in terms
of AB
to circle F. Determine
afi ACB
mZG in terms of 2n
alone and
.
Z
Angles Associated with a Circle
FB L BG; FA L
/.A and /.8
AG
are right
(A tangent line is perpendicular to the
radius at the point of tangency.)
angles
mZA: mZB :90o
mZF
(Definition of perpendicular lines)
(De/inition of right angles)
: *78
(A central angle has a measure equal to
that of its intercepted arc.)
mZA+mZF +mZG+mZB
:360"
(The sum of the measures of the angles in a
quadrilateral is 360' .)
90o + mZ_G+
g0" + mIE
: 360"
(Substitution Property)
(subtraction)
mlG = lg0" -*78
This leads to the conclusion
that ZG and ZF are supplementary.
In addition,
*78 :
mzG
:
360,
180'
(The sum of the arcs in a circle is
-*frfi
-(\SoO" -
360' .)
*m\ /
(Substitution)
mzG:*frd*tBT'
*frE 2.mZ.G = *frfi - *78
mlG +mZG :
mZG
180'
l::-(nACB-wAB)
2'
+ lB0" -*78
(Addition)
(Division)
V
Trainer/Instructor Notes: Polygons &
@ dfr is tangent to circle F
ZBCD and IBAC .
Circles
Angles Associated with a Circle
at C. Determine the relationship between the measures
of
FC LCD
(A tangent line is perpendicular
to the radius at the point of
tangency.)
ZFCD
(D efi niti on
is a right angle
of p e rp e ndi cul or
lines)
nZFCD: 90'
(Definition of right angle)
nZFCD:wZFCB+wZBCD
(Angle Addition)
nZFCB+/BCD:90o
(Substiturion)
nZFCB :90o -nZBCD
(Subtraction)
fC=f
(All radii in a circle are
congruent.)
Z.FBC = ZFCB
(If two sides of a triangle are
congruent, then the angles
opposite those sides are
congruent.)
mZFBC
:90' -w/-BCD
MZFBC + MZFCB + MICFB
(Substitution)
: ]8OO
(The sum of the meosures of the
angles in a triangle
90'
- mlBCD + 90' -
wZCFB = 2 .nZBCD
ruZBCD + nZCFB
: l 80o
(Substitution)
(Subtraction)
is
180" .)
Trainer/Instructor Notes: Polygons &
Ciicles
Angles Associated with a Circle
(A central angle has a measure
equal to that of its intercepted
arc.)
m/.CFB = mBC
l^
(The measure of an inscribed
angle is half of the measure of
its intercepted arc.)
nZBAC =1-mBC
2
mBC = 2.mlBAC
(Multiplication)
2.mlBAC:2.m/.BCD
(Substitution)
m/.BAC:wZBCD
(Division)
In addition, we can see that m/-BCD
:
t^
!-.mBC
.
2
@) AC l. tangent to circle D
intercepted arcs
AB
at B. Determine the measur e
ZABG = IACB
in terms of the
(The angle between a tangent and a
chord is congruent to the inscribed
angle intercepted by the arc on the
same side as the tangent.)
1^
wIACB = -mAB andmlBAC = -mBC
wIBAC=mZG+m/.ABG
lG
G
and CB.
1^
22
of
(The measure of an inscribed angle
is equal to half of the measure of its
intercepted arc.)
(The measure of the exterior angle
of a triangle is equal to the sum of
the measures of the interior remote
angles.)
Angles Associated with a
nZBAC=mlG+nZACB
1-l^
C,Iqg
(Substitution)
(Substitution)
-mBC=mZG+-mAB
))
mzG
-*h)
=l*fu
2\
2
2 -1*fr:l(*6i
(Subtraction)
lT