Download Chapter 4 Proof #1

Chapter 4 Proof Practice
This is to give students more practice with proofs.
1) Print out the proofs on heavy paper or cardstock and put in a paper protector
2) Print out the justifications and statements on cardstock. (I have done statements on one color, justifications on another, or make it a
little more challenging and have them all the same color – students should be able to figure out which is which).
3) Have students work on the proofs in groups or pairs. (I have several of each proof printed so there is enough for the whole class to
work in pairs).
4) Either have students write down the proof once they are done or just check it off in some way so you know which ones they have
completed.
5) Occasionally I had students waiting over another students shoulder for a particular number towards the end. But generally students
were engaged and on task.
6) I ended up taking out the triangle congruence postulates as I expected students to know which one it was, you can do it either way.
Also, you could give them just the statements or just the justifications to make it more challenging.
Chapter 4 Proof #1
Given: IE  GH ; EF  HF ; F is the midpoint of GI
Prove: EFI  HFG
Proof 1 Statements
Proof 1 Justifications
IE  GH
Given
EF  HF
Given
F is the midpoint of GI
Given
FG  FI
EFI  HFG
Proof 2 statements
Defintion of a midpoint
A point is the midpoint if and only if it divides the segment
into 2 equal parts.
SSS Postulate
If the three sides of one triangle are congruent to the three
sides of another triangle, then the triangles are congruent.
proof 2 justifications
WZ  ZS
Given
SD  DW
Given
ZD  ZD
Reflexive Property of Congruence
WZD  SDZ
SSS Postulate
If the three sides of one triangle are congruent to the three
sides of another triangle, then the triangles are congruent.
Chapter 4 Proof #2
Given: WZ  ZS  SD  DW
Prove: WZD  SDZ
Chapter 4 Proof #3
Given: AB  CD , BAC  DCA
Prove: ABC  CDA
Proof 3 Statements
Proof 3 Justifications
AB  CD
Given
BAC  DCA
Given
AC  AC
ABC  CDA
Proof 4 Statements
X is the midpoint of AG and NR
AX  XG
NX  XR
Reflexive Property of Congruence
SAS Postulate
If the two sides and the included angle of one triangle are
congruent to the two sides and the included angle of
another triangle, then the triangles are congruent.
Proof 4 justifications
Given
Defintion of a midpoint
A point is the midpoint if and only if it divides the segment
into 2 equal parts.
Defintion of a midpoint
A point is the midpoint if and only if it divides the segment
into 2 equal parts.
AXN  GXR
Vertical Angles Theorem
Vertical Angles are congruent
AXN  GXR
If the two sides and the included angle of one triangle are
congruent to the two sides and the included angle of
another triangle, then the triangles are congruent.
SAS Postulate
Chapter 4 Proof #4
Given: X is the midpoint of AG and NR
Prove: AXN  GXR
Chapter 4 Proof #5
Given: GK bisects JGM , GJ  GM
Prove: GJK  GMK
Proof #5 Statements
Proof #5 Justifications
GJ  GM
Given
GK bisects JGM
Given
GK  GK
GJK  GMK
JGK  MGK
Proof #6 Statements
Reflexive Property of Congruence
SAS Postulate
If the two sides and the included angle of one triangle are
congruent to the two sides and the included angle of
another triangle, then the triangles are congruent.
Definition of an Angle Bisector
A segment is the angle bisector if and only if it
divides an angle into 2 congruent angles.
Proof #6 Justifications
AC  CE
Definition of an Segment Bisector
A segment is the segment bisector if and only
if it divides a segment into 2  segments.
AE and BD bisect each other
Given
CD  CB
Definition of an Segment Bisector
A segment is the segment bisector if and only
if it divides a segment into 2  segments.
SAS Postulate
ACB  ECD
If the two sides and the included angle of one triangle are
congruent to the two sides and the included angle of
another triangle, then the triangles are congruent.
ACB  ECD
Vertical Angles Theorem
Vertical Angles are congruent
Chapter 4 Proof #6
Given: AE and BD bisect each other
Prove: ACB  ECD
Chapter 4 Proof #7
Given: FG || KL ; FG  KL
Prove: FGK  KLF
Proof #7 Statements
Proof #7 Justifications
FG || KL
Given
FG  KL
Given
FK  FK
FGK  KLF
GFK  LKF
Proof #8 Statements
Reflexive Property of Congruence
SAS Postulate
If the two sides and the included angle of one triangle are
congruent to the two sides and the included angle of
another triangle, then the triangles are congruent.
Alternate Interior Angles Theorem
If two parallel lines are intersected by a transversal
then alternate interior angles are congruent.
Proof #8 Justifications
BAC  DAC
Given
AC  BD
Given
AC  AC
ABC  ADC
Reflexive Property of Congruence
ASA Postulate
If the two angles and the included side of one triangle are
congruent to the two angles and the included side of
another triangle, then the triangles are congruent.
BCA  DCA
All Right Angles are Congruent
BCA and DCA are right angles
Definition of Perpendicular Lines
Two segments are perpendicular if and only if
they intersect to form 4 right angles.
Chapter 4 Proof #8
Given:
BAC  DAC
AC  BD
Prove: ABC  ADC
Chapter 4 Proof #9
Given:
V  Y
WZ bisects VWY
Prove: VWZ  YWZ
Proof #9 Statements
Proof #9 Justifications
V  Y
Given
WZ bisects VWY
Given
WZ  WZ
VWZ  YWZ
VWZ  YWZ
Proof #10 Statements
Reflexive Property of Congruence
AAS Theorem
If the two angles and a non-included side of one triangle
are congruent to the two angles and a non-included side of
another triangle, then the triangles are congruent.
Definition of an Angle Bisector
A segment is the angle bisector if and only if it
divides an angle into 2 congruent angles.
Proof #10 Justifications
PQ  QS
Given
RS  QS
Given
T is the midpoint of PR
PQT  RST
PQT  RST
PQT is a right angle
RST is a right angle
PT  TR
QTP  STR
Given
AAS Theorem
If the two angles and a non-included side of one triangle
are congruent to the two angles and a non-included side of
another triangle, then the triangles are congruent.
All Right Angles are Congruent
Definition of Perpendicular Lines
Two segments are perpendicular if and only if
they intersect to form 4 right angles.
Definition of Perpendicular Lines
Two segments are perpendicular if and only if
they intersect to form 4 right angles.
Defintion of a midpoint
A point is the midpoint if and only if it divides the segment
into 2 equal parts.
Vertical Angles Theorem
Vertical angles are congruent
Chapter 4 Proof #10
Given:
PQ  QS; RS  QS,
T is the midpoint of PR
Prove: PQT  RST
Chapter 4 Proof #11
Given: N  P ; MO  QO
Prove: MON  QOP
Proof #11
N  P
Given
MO  QO
Given
MON  QOP
MON  QOP
Vertical Angles are congruent
AAS Theorem
If the two angles and a non-included side of one triangle
are congruent to the two angles and a non-included side of
another triangle, then the triangles are congruent
Proof #12
E  D
Given
AE  BD
Given
AE || BD
Given
AEB  BDC
If the two angles and the included side of one triangle are
congruent to the two angles and the included side of
another triangle, then the triangles are congruent.
EAB  DBC
ASA Postulate
Corresponding Angles Postulate
If two parallel lines are intersected by a transversal then
corresponding angles are congruent.
Proof #13
DH bisects BDF
Given
1  2
Given
DH  DH
Reflexive Property of Congruence
BDH  FDH
If the two angles and the included side of one triangle are
congruent to the two angles and the included side of
another triangle, then the triangles are congruent.
BDH  FDH
ASA Postulate
Definition of an Angle Bisector
A segment is the angle bisector if and only if it divides the
angle into two congruent angles.
Proof #12
Given: E  D
AE  BD
AE || BD
Prove: AEB  BDC
Proof #13
Given: DH bisects BDF
1  2
Prove: BDH  FDH