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Geometry Proofs: Chapter 8, section 8.2
Theorem 8.3: If a quadrilateral is a parallelogram, then its opposite
sides are congruent.
Given: ABCD.
Prove: AB CD# ; AD BC#
Statements Reason
1. ABCD. 1. Given
2. Draw AC 2.
3. 3. Def. of .
4. 4. Alternate Interior Angles Th.
5. AC AC# 5.
6. ABC CDA#++ 6.
7. 7. CPCTC
Theorem 8.4: If a quadrilateral is a parallelogram, then its opposite
angles are congruent.
Given: ABCD.
Prove: A C� # � ; B D� # �
Statements Reasons
1. ABCD. 1. ABCD.
2. 2. 2 pts determine a line
3. 3. Def. of .
4. 4. Alternate Interior Angles Th.
5 . m BAC m DCA
m DAC m BCA
� �� �
5.
6 . m BAC m DAC m DCA m BCA� � � � � � 6.
7 . m BAD m BAC m DAC
m DCB m DCA m BCA
� � � �� � � �
7.
8 . m BAD m DCB� � 8.
9 . BAD DCB� # � 9.
10. AC AC# 10.
11. 11.
12. B D� # � 12.
D
B
C
A
B
C D
A
Theorem 8.5: If a quadrilateral is a parallelogram, then its
consecutive angles are supplementary.
Given: ABCD.
Prove: A� is supplementary to B� ; B� is supplementary to C� ;
C� is supplementary to D� ; D� is supplementary to A�
Statements Reasons
1. ABCD. 1.
2. AB DC& 2.
3. 3. Consecutive Interior Angles Th.
4. AD BC& 4.
5. 5. Consecutive Interior Angles Th.
Theorem 8.6: If a quadrilateral is a parallelogram, then its diagonals
bisect each other.
Given: ABCD.
Prove: bisects ; bisects AC BD BD AC
Statements Reason
1. ABCD. 1.
2. 2. 2 points determine a line
3. 3. If two lines intersect, then they
intersect at a point.
4. ; AB DC AD BC& & 4.
5. 5. Alternate Interior Angles Th.
6. AC AC# 6.
7. ABC CDA#+ + 7.
8. AB DC# 8.
9. 9. Vertical Angles Congruence Th.
10. ABE CDE#++ 10.
11. 11. CPCTC
12. 12. Definition of Segment Bisector
D
B
C
A
D
B
C
A
Geometry Proofs: Chapter 8, section 8.3
Theorem 8.7: If both pa irs of opposite sides of a quadrilatera l are congruent, then the quadrilatera l is a parallelogram. Given: Prove: Statements: Reasons:
1. 1.
2. 2. Two points determine a line
3. 3.
4. 4.
5. 5.
6. 6.
7. 7.
Theorem 8.8: If both pa irs of opposite angles of a quad are congruent, then the quad is a parallelogram. Given: Prove: Statements: Reasons:
1. 1. Given
2. 2.
3. 3. Quad. Sum Theorem
4a. 4b.
4.
5a. 5b.
5. Distributive Prop.
6a. 6b.
6.
7a. 7b.
7.
8.
8.
9. 9. Def. of Parallelogram
, AB DC AD BC# #ABCD.
, AB DC AD BC# #
AC AC#
,BAC DCA� # �BCA DAC� # �
ABCD.
, A C B D� # � � # �ABCD.
= , =m A m C m B m D� � � �
360m A m B m A m B� � � � � � �
360m A m D m A m D� � � � � � �
180m A m B� � � 180m A m D� � �
, AD BC AB DC& &
Geometry Proofs: Chapter 8, section 8.3
Theorem 8.9: If one pa ir of opposite sides of a quadrilateral are congruent and paralle l, then the quadrilateral is a parallelogram. Given: Prove:
Statements: Reasons: 1. 1. Given
2. Draw AC 2.
3. 3. Alternate Interior Angles
Theorem 4. 4.
5. BAC DCA#+ + 5.
6. 6.
7. 7. Alternate Interior Angles
Converse Th. 8. 8.
Theorem 8.10: If the diagona ls of a quadrilateral bisect each other, then the quadrilatera l is a parallelogram. Given: Prove:
Statements: Reasons: 1.
1. Given
2.
2.
3.
3. Vert. Angle Congruence Theorem
4.
4.
5.
5.
6.
6.
7.
7. Def. of Parallelogram
, AB DC AB DC#&ABCD.
BCA DAC� # �
ABCD.
bisects AC BD
ABCD.
,DE BE# AE CE#
,AED CEB#+ +AEB CED#+ +
,AD BC& AB DC&
bisects BD AC
Geometry Proofs: Chapter 8, Section 8.4 Theorem 8.11: A parallelogram is a rhombus if and only if its diagonals are perpendicular.
Given:Rhombus ABCD,
ABCD. Given: , ABCD AC BDA.
Prove: AC BDA Prove: Rhombus ABCD
Statements Reasons
1. , ABCD AC BDA.
1.Given
2. bisects BD AC
2.
3. 3.Def. of Segment Bisector
4. & BEC are Rt. 'BEA s� � � 4.
5. BEA BEC� # � 5.
6. 6. Reflexive Prop.
7. ABE CBE#+ + 7.
8. AB BC# 8.
9. 9.
If , then 2 pairs of opp. sides #.
10. AB BC CD AD# # # 10. Transitive prop. # (8,9)
11. 11.
Statements Reasons
1. Rhombus ABCD
1. Given
2.
2. Def. of Rhombus
3. Parallelogram ABCD 3.
4. bisects BD AC 4.
5. 5. Def. of Segment Bisector
6.BE BE# 6.
7. ABE CBE#+ + 7.
8. 8. CPCTC
9. 9. Def. of Linear Pair
10. 10.
D
B C
A
D
B C
A
E E
Geometry Proofs: Chapter 8, Section 8.4 Theorem 8.12: A parallelogram is a rhombus if and only if its diagonals bisect a pair of opposite angles.
Given: Rhombus ABCD, ABCD. Given: ; bisects & ,
bisects &
ABCD AC DAB BCD
BD ABC CDA
� �
� �
.
Prove: bisects & ,
bisects &
AC DAB BCD
BD ABC CDA
� �
� � Prove: Rhombus ABCD
Statements Reasons
1. ; bisects & ,
bisects &
ABCD AC DAB BCD
BD ABC CDA
� �
� �
. 1.
2 ; ,
;
DAC BAC BCA DCA
ABD CBD CDB ADB
� # � � # �� # � � # �
2.
3. ;AC AC BD BD# # 3.
4. , ABC ADC ABD CBD# #+ + + + 4.
5. ; ;AB AD BC CD AB BC# # # 5.
6. AB BC CD AD# # # 6.
7. Rhombus ABCD 7.
Statements Reasons
1. 1.
2. , ; ,AB AD BC CD AB BC AD CD# # # # 2.
3. ;AC AC BD BD# # 3.
4. ; ABC ADC ABD CBD# #+ + + + 4.
5.; ,
;
DAC BAC BCA DCA
ABD CBD CDB ADB
� # � � # �� # � � # �
5.
6. bisects & ,
bisects &
AC DAB BCD
BD ABC CDA
� �
� �
6.
D
B C
A
D
B C
A
Geometry Proofs: Chapter 8, Section 8.4 Theorem 8.13: A parallelogram is a rectangle if and only if its diagonals are congruent.
Given: Rectangle ABCD, ABCD. Given: ; ABCD AC BD#.
Prove: AC BD# Prove: Rectangle ABCD
Statements Reasons
1. Rectangle ABCD, ABCD. 1.
2. AD BC# 2.
3. & are Right 'sADC BCD� � � 3.
4. ADC BCD� # � 4.
5. 5.
6. ADC BCD#+ + 6.
7. 7.
Statements Reasons
1. ; ABCD AC BD#. 1.Given
2. 2.
If , then 2 pairs of opp. sides #.
3. AD AD# 3.
4. BAD CDA#+ + 4.
5. BAD CDA� # � 5.
6.m BAD m CDA� � 6.
7.
7. If , then consec. 's are suppl.�.
8. 180m BAD m CDA� � � q 8.
9. 180m BAD m BAD� � � q 9.
10. 2( ) 180m BAD� q 10.
11. 11.
12. 90m CDA� q 12.
13. 13. If , then 2 pairs opp 's � #.
14. 14. Def. of 's# �
15. 90 ; 90m BCD m ABC� q � q 15.
16. , , , are Rt. 'BAD ABC BCD CDA s� � � � � 16.
17. Rectangle ABCD 17.
D C
B A
D C
B A
Theorem 8.14: If a t r a pezoid is isosceles , t hen both p airs of b ase a ngles a re congr uen t G iven: Isosceles Tra pezoid A B C D
Prove: :A D B BCD� # � � # �
Theorem 8.15: If t r apezoid h as a pa irs of congr uen t b ase a ngels , t hen i t is a n isosceles tr apezoid Given: Trapezoid ABCD; A D� # � Prove: Isosceles Trapezoid ABCD
Sta temen ts Reasons
1 . Tra pezoid A B C D ; A D� # � 1 .
2 . BC AD& 2 .
3 . Draw CE AB& 3 .
4 . ABCE. 4 .
5 . 5 . If p a ra l lelogra m , t hen 2 p a irs of opposite sides a re congr uen t
6 . A CED� # � 6 .
7 . CED D� # � 7 .
8 . 8 . Converse of B ase A ngles Th .
9 . 9 . Tra nsi tive Property of Congr uence(#5 ,8)
1 0 . Isosceles Trapezoid ABCD 1 0 .
Sta temen ts Reasons
1 . Isosceles Tra pezoid A B C D 1 .
2 . 2 . D ef i n i t ion of Isosceles Tra pezoid
3 . Draw CE AB& 3 .
4 . ABCE. 4 .
5 . 5 . If p a ra l lelogra m , t hen 2 p a irs of opposite sides a re congr uen t
6 . CE CD# 6 .
7 . CED D� # � 7 .
8 . � # �A CED 8 .
9 . 9 . Tra nsi tive Property of Congr uence(#7 ,8)
1 0 . 1 0 . Consec u tive In terior A ngles T heorem
1 1 . 1 1 . Congr uen t S u pp lemen ts T heorem
Geometry Proofs: Chapter 8, Sect ion 8.5 (Part A)
A
C B
D A
B C
D
Theorem 8.16: A tra pezoid is isosceles if a n d on ly if i ts d iagon a ls a re congr uen t . Given: Isosceles Trapezoid ABCD Prove: AC BD#
Given: Trapezoid ABCD: AC BD# Prove: Isosceles Tra pezoid A B C D
Sta temen ts Reasons
1 . 1 . Given
2 . ;BC AD AB CD#& 2 .
3 . 3 .If isosceles tr apezoid , t hen b ase a ngles a re congr uen t
4 . AD AD# 4 .
5 . ABD DCA' # ' 5 .
6 . AC BD# 6 .
Sta temen ts Reasons
1 .Tra pezoid A B C D; AC BD# 1 .
2 . BC AD& 2 .
3 . Draw &BX AD CY ADA A 3 .
4 . ;BX BC CY BCA A 4 .
5 . 5 . D ef i n i t ion of Perpen d ic ula r L i nes
6 . Rectanlgle BCYX 6 .
7 . BCYX. 7 .
8 . 8 . If p a ra l lelogra m , t hen 2 pa irs of opposite sides a re congr uen t
9 . and
are right triangleBXD CYA' '
9 .
1 0 . BXD CYA' # ' 1 0 .
1 1 . XDB YAC� #� 1 1 .
1 2 . 1 2 . Reflex ive Property of Congr uence
1 3 . ACD DBA' # ' 1 3 .
1 4 . CD BA# 1 4 .
1 5 .Isosceles Tra pezoid A B C D 1 5 .
A
B C
D
A
B C
D
Geometry Proofs: Chapter 8, Section 8.5 (Part B)
Theorem #8.17-Midsegment Theorem for Trapezoids: The midsegment Theorem #8.18: If a quadrilateral is a kite, then its diagonals are perpendicular. of a trapezoid is parallel to each base and its length is one half the sum of the lengths of the bases.
Statements Reasons 1. Trapezoid with Midsegment ABCD MN 1.
2. 2. If 2 lines intersect, thenthey intersect at a point.
M and N are the midpoints of 3. &BC AD 3.
4. 4. Definition of a Midpoint
5. 5. Definition of a Trapezoid
6. ABN GCN� # � 6. 7. ANB GNC� # � 7. 8. 8. ASA-Congruence Postulate
9. 9. CPCTC
10. is the Midpoint of N AG 10.
11. 11. Definition of a Midsegmentfor Triangles
12. MN CD& 12.
13. MN AB& 13.
114. 2
MN DG 14.
15. DG CG CD � 15.
16. CG AB# 16.
17. CG AB 17. 18. DG AB CD � 18.
119. ( )2
MN AB CD � 19.
Statements Reasons
1. Kite ABCD 1.
2. 2. Definition of a Kite
3. AC AC# 3.
4. ABC ADC#+ + 4.
5. 5. CPCTC
6. AE AE# 6.
7. 7. SAS-Congruence Theorem
8. AEB AED� # � 8.
9. 9. Definition of a Linear Pair
10. AC BDA 10.
Given: Trapezoid with Midsegment 1Prove: ; ; ( )2
ABCD MN
MN AB MN CD MN AB CD �& &
Given: Kite
Prove:
ABCD
AC BDA
Theorem 8.19: If a quadrilateral is a kite, then exactly one pair of opposite angles are congruent. Part 1
Part 2: (Show is not indirectly)A C� # �
Statements Reasons 1. Kite ABCD 1.
2. , AB AD CB CD# # 2.
3. 3. 2 points determine a line
4. AC AC# 4.
5. 5. SSS-Congruence Postulate
6. 6. CPCTC
Statements Reasons
1. A C� # � 1.
2. B D� # � 2.
3. ABCD. 3.
4. 4.
5. 5. Given
6. 6. Definition of a Kite
Given: Kite Prove:
ABCDB D� # �
Given: Kite , Prove: is notAssume: _______________
ABCD B DA C
� # �� # �
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