Unit D - Circles 6.4 Circle Proofs Date: ______Period: ______

Geometry H Name: ______Unit D - Circles 6.4 Circle Proofs 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, andOR CD Prove: OR bisects CD

Statements Reasons

1. Circle O with a chord radii 1. Given 2. All radii of the same and circle are congruent 2. OC OD 3. Definition of isosceles 3. COD is isosceles 4. Base ’s of an 4. CD isosceles are 5. ORC, ORD are right ’s 5. Definition of ORC ORD 6. 6. All right ’s are 7. ORC ORD 7. AAS congruence conj. 8. CR RD 8. CPCTC

9. bisects 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 E A AD together form , so together they measure 180º since the C measure of a semicircle = 180 . mBA 108 , so 36 D mAD 180 108; mAD 72 . This means that mB 36 , since the measure of an inscribed angle is ½ the measure of its intercepted arc. BD since they both measure 36º, so by the converse of the parallel lines conjecture (AIA). 3. Given: AC tangent to circle D at A and circle P at C Prove: DP   (Hint: you do NOT have to prove C congruent triangles! That’s not possible.) D

B P

Statements Reasons

1. tangent to circle D at A and 1. Given

circle P at C

2. AD AC , PC AC 2. Tangent are to radii

3. AC, are right ’s 3. Definition of 4. All right ’s are 4. AC 5. Vertical ’s are 5. DBA PBC 6. 3rd angle conjecture 6.

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 A Prove: ACD   ABD

Statements Reasons C O 1. Circle O with inscribed angles 1. Given and 2. Measure of inscribed is D B 1 1 half the measure of the 2. m C mAD , m B mAD 2 2 intercepted arc 3. m C m B 3. Substitution 4. CB 4. Definition of ’s

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

A B 1. Circle O with diameter and 1. Given O inscribed in a semicircle 2. Measure of inscribed is 1 half the measure of the 2. m C mADB 2 intercepted arc D 3. The measure of a 3. mADB 180 semicircle = 180  1 4. mC 180 4. Substitution 2 mC 90 5. Multiplication 5. 6. C is a right angle 6. Definition of right angle

6. Prove the Cyclic Quadrilateral Conjecture: The opposite angles of a cyclic quadrilateral are supplementary.

Given: Circle O with inscribed quadrilateral LICY Y L C Prove: and are supplementary L

Statements Reasons O

1. Circle O with inscribed 1. Given C I quadrilateral LICY 2. The measure of a circle = 2. mYCI mILY 360 360 (and arc addition) 1 1 3. Measure of inscribed is 3. m L mYCI , m C mILY 2 2 half the measure of the intercepted arc 4. 2m L mYCI , 2m C mILY 4. Multiplication prop of = 5. 2m L 2 m C 360 5. Substitution 6. m L m C 180 6. Division prop of = 7. and are supplementary 7. Definition of supplementary

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 A B Prove: BC DA O

D C

Statements Reasons

1. Circle O with chord and 1. Given

2. m BDC m ABD 2. Parallel lines conjecture

1 1 (AIA) 3. m BDC mBC , m ABD mDA 2 2 3. Measure of inscribed is

11 half the measure of the 4. mBC mDA 22 intercepted arc 4. Substitution 5. mBC mDA 5. Multiplication prop of = 6. 6. Definition of arcs

E 8. Given: Circle O with inscribed trapezoid GATE Prove: GATE is an isosceles trapezoid G

Statements Reasons O T 1. Circle O with inscribed 1. Given A trapezoid GATE 2. Definition of trapezoid

3. Parallel lines intercept 2. GE AT congruent arcs on a circle 3. GA ET 4. Congruent arcs intercept 4. GA ET congruent chords in a circle 5. GATE is an isosceles 5. Definition of isosceles trapezoid trapezoid

A 9. Given: Circle O with AE BE B AC BD Prove: E C Statements Reasons O D

1. Circle O with 1. Given 2. AB 2. Inscribed angles that intercept the same arc are congruent. 3. AEC BED 3. Vertical ’s are 4. AEC BED 4. ASA congruence conj. 5. AC BD 5. CPCTC 6. Congruent chords intercept 6. 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) D

G Statements Reasons

OA 1. Circle A with inscribed 1. Given L parallelogram GOLD with 2. Definition of parallelogram and 3. opposite angles of a cyclic O 2. GOLD is a quadrilateral quadrilateral are 3. DO, are supplementary, supplementary LG, are supplementary 4. Definition of m D m O 180 4. , supplementary m L m G 180 5. Opposite ’s in a 5. m D m O , m L m G parallelogram are = in 6. m D m D 180 , measure m O m O 180 , m L m L 180 , 6. Substitution

m G m G 180 7. Combine like terms 2mD 180 , 2mO 180 , 7. 8. Division prop of = 2mL 180 , 2mG 180 8. mD 90 , mO 90 , 9. Definition of equiangular mL 90 , mG 90 9. GOLD is equiangular 10. Definition of rectangle 10. GOLD is a rectangle

11. Given: Circle O with chords AB AC Prove: AMON is a kite

B Statements Reasons

1. Given 1. Circle O with chords M 2. Given O 2. AB MO , AC NO C 3. Definition of segments 3. mAB mAC 4. from center of circle to N 4. OM bisects AB ,ON bisects AC chord bisects the chord A 1 1 5. mAM mAB , mAN mAC 5. Definition of bisect 2 2

1 6. Substitution 6. mAM mAC 2 7. Substitution 8. Definition of segments 7. mAM mAN 9. Congruent chords are 8. AM AN equidistant to center of 9. OM ON circle 10. AMON is a kite 10. Definition of kite

12. Given: Circle O with diameter AB and chord AD , and OE AD Prove: BE DE A

Statements Reasons

1. Circle O with diameter and 1. Given O chord , and 2. Parallel lines conj (AIA) 2. m BOE m BAD 3. Arc measure definition 3. m BOE mBE D 4. Measure of inscribed is B 1 E 4. m BAD mBD half the measure of the 2 intercepted arc 1 5. mBE mBD 5. Substitution 2

6. 2mBE mBD 6. Multiplication prop of = 7. mBD mBE mDE 7. Arc addition 8. 2mBE mBE mDE 8. Substitution 9. Subtraction prop of = 9. mBE mDE 10. Definition of arcs 10.