Geometry H
Unit D - Circles
6.4 Circle Proofs
Name: _________________________________
Date: _________________ Period: _______
Note: In your proofs, you are allowed to use the following:
the measure of a semicircle = 180  ; the measure of a circle = 360 
all radii of the same circle are congruent
1. Prove the Perpendicular to a Chord Conjecture: The perpendicular from the center of a
circle to a chord is the bisector of the chord.
Given: Circle O with a chord CD, radii OC , OD, and OR  CD
Prove: OR bisects CD
Statements
Reasons
1. Circle O with a chord CD, radii
OC , OD, and OR  CD
2. OC OD
3.
COD is isosceles
4.
C
D
5.
ORC, ORD are right
6.
ORC
ORD
7.
ORC
ORD
8. CR
9.
’s
RD
OR bisects CD
1. Given
2. All radii of the same
circle are congruent
3. Definition of isosceles
4. Base ’s of an
isosceles are
5. Definition of
6. All right ’s are
7. AAS congruence conj.
8. CPCTC
9. Definition of bisect
2. Given: Circle E with diameter DB, inscribed angles CDB and DBA, mBDC  36,
and mAB  108
Explain why AB DC (you do not need to do a two-column
108
B
proof.)
DB is a diameter, which means BAD is a semicircle. Arcs BA and
AD together form BAD , so together they measure 180º since the
measure of a semicircle = 180  . mBA
mAD
180 108; mAD
C
108 , so
72 . This means that m B
A
E
36
D
36 , since the
measure of an inscribed angle is ½ the measure of its intercepted arc.
B
D since they both
measure 36º, so AB DC by the converse of the parallel lines conjecture (AIA).
3. Given: AC tangent to circle D at A and circle P at C
Prove: D  P (Hint: you do NOT have to prove
congruent triangles! That’s not possible.)
C
D
B
P
A
Statements
Reasons
1. AC tangent to circle D at A and
circle P at C
1. Given
2. AD AC , PC AC
3.
A, C are right ’s
2.
3.
4.
5.
6.
4.
A
5.
DBA
C
PBC
6. D  P
Tangent are to radii
Definition of
All right ’s are
Vertical
’s are
3rd angle conjecture
4. Prove the Inscribed Angles Intercepting the Same Arc Conjecture: Inscribed angles that
intercept the same arc are congruent.
Given: Circle O with inscribed angles ACD and ABD
Prove: ACD  ABD
Statements
1. Circle O with inscribed angles
ACD and ABD
1
1
2. m C
mAD , m B
mAD
2
2
3. m C m B
C
B
4.
Reasons
1. Given
2. Measure of inscribed is
half the measure of the
intercepted arc
3. Substitution
4. Definition of
’s
A
C
O
D
B
5. Prove the Angles Inscribed in a Semicircle Conjecture: Angles inscribed in a semicircle
are right angles.
Given: Circle O with diameter AB and ACB inscribed in a semicircle
Prove: ACB is a right angle
C
Statements
Reasons
1. Circle O with diameter AB and
ACB inscribed in a semicircle
1
2. m C
mADB
2
1. Given
2. Measure of inscribed is
half the measure of the
intercepted arc
3. The measure of a
semicircle = 180 
4. Substitution
5. Multiplication
6. Definition of right angle
3. mADB
180
1
4. m C
180
2
5. m C 90
6.
C is a right angle
A
B
O
D
6. Prove the Cyclic Quadrilateral Conjecture: The opposite angles of a cyclic quadrilateral
are supplementary.
Given: Circle O with inscribed quadrilateral LICY
Prove: L and C are supplementary
Statements
3. m L
4.
5.
6.
7.
mILY 360
1
mYCI , m C
2
L
Reasons
1. Circle O with inscribed
quadrilateral LICY
2. mYCI
Y
1
mILY
2
2m L mYCI , 2m C mILY
2m L 2m C 360
m L m C 180
L and C are supplementary
1. Given
2. The measure of a circle =
360  (and arc addition)
3. Measure of inscribed is
half the measure of the
intercepted arc
4. Multiplication prop of =
5. Substitution
6. Division prop of =
7. Definition of supplementary
O
I
C
7. Prove the Parallel Lines Intercepted Arcs Conjecture: Parallel lines intercept congruent
arcs on a circle.
Given: Circle O with chord BD and AB CD
B
A
Prove: BC  DA
O
D
Statements
Reasons
1. Circle O with chord BD and AB CD
2. m BDC
m ABD
1
3. m BDC
mBC , m ABD
2
1
1
4.
mBC
mDA
2
2
5. mBC
mDA
6. BC  DA
1
mDA
2
1. Given
2. Parallel lines conjecture
(AIA)
3. Measure of inscribed is
half the measure of the
intercepted arc
4. Substitution
5. Multiplication prop of =
6. Definition of arcs
8. Given: Circle O with inscribed trapezoid GATE
Prove: GATE is an isosceles trapezoid
Statements
1. Circle O with inscribed
trapezoid GATE
2. GE AT
3. GA
ET
4. GA ET
5. GATE is an isosceles
trapezoid
C
E
G
Reasons
1. Given
2. Definition of trapezoid
3. Parallel lines intercept
congruent arcs on a circle
4. Congruent arcs intercept
congruent chords in a circle
5. Definition of isosceles
trapezoid
O
A
T
A
9. Given: Circle O with AE  BE
Prove: AC
B
BD
E
C
Statements
1. Circle O with AE  BE
2. A
B
3.
4.
AEC
AEC
5. AC
BD
6. AC
BD
BED
BED
O
Reasons
D
1. Given
2. Inscribed angles that intercept the
same arc are congruent.
3. Vertical
’s are
4. ASA congruence conj.
5. CPCTC
6. Congruent chords intercept
congruent arcs in a circle
10. Given: Circle A with inscribed parallelogram GOLD with GD OL and GO DL
Prove: GOLD is a rectangle (i.e. GOLD is equiangular)
Statements
1. Circle A with inscribed
parallelogram GOLD with
GD OL and GO DL
2. GOLD is a quadrilateral
3.
D, O are supplementary,
L, G are supplementary
4. m D m O 180 ,
m L m G 180
5. m D m O , m L m G
6. m D m D 180 ,
m O m O 180 ,
m L m L 180 ,
m G m G 180
7. 2m D 180 , 2m O 180 ,
2m L 180 , 2m G 180
8. m D 90 , m O 90 ,
m L 90 , m G 90
9. GOLD is equiangular
10. GOLD is a rectangle
Reasons
1. Given
2. Definition of
parallelogram
3. opposite angles of a cyclic
quadrilateral are
supplementary
4. Definition of
supplementary
5. Opposite ’s in a
parallelogram are = in
measure
6. Substitution
7. Combine like terms
8. Division prop of =
9. Definition of equiangular
10. Definition of rectangle
D
G
OA
L
O
11. Given: Circle O with chords AB  AC
Prove: AMON is a kite
B
Statements
1. Circle O with chords AB  AC
2. AB
MO , AC
NO
3. mAB  mAC
4. OM bisects AB , ON bisects AC
1
1
5. mAM
mAB , mAN
mAC
2
2
1
6. mAM
mAC
2
7. mAM
mAN
8. AM
Reasons
1. Given
2. Given
3. Definition of
4.
5.
6.
7.
8.
M
O
C
segments
from center of circle to
chord bisects the chord
Definition of bisect
Substitution
Substitution
Definition of
segments
N
A
9. Congruent chords are
equidistant to center of
circle
10. Definition of kite
AN
9. OM ON
10. AMON is a kite
12. Given: Circle O with diameter AB and chord AD , and OE AD
Prove: BE
DE
A
Statements
1. Circle O with diameter AB and
chord AD , and OE AD
2. m BOE
m BAD
3. m BOE
mBE
1
mBD
2
4. m BAD
5. mBE
1
mBD
2
6. 2mBE
mBD
7. mBD
mBE
8. 2mBE
9. mBE
10. BE
mBE
mDE
DE
mDE
mDE
Reasons
1. Given
2. Parallel lines conj (AIA)
3. Arc measure definition
4. Measure of inscribed is
half the measure of the
intercepted arc
5. Substitution
6. Multiplication prop of =
7. Arc addition
8. Substitution
9. Subtraction prop of =
10. Definition of arcs
O
D
B
E