Transcript 2.5 Proving Statements about Line Segments

Theorems are statements that can be
proved
Theorem 2.1 Properties of Segment Congruence
Reflexive AB ≌ AB
All shapes are ≌ to them self
Symmetric If AB ≌ CD, then CD ≌ AB
Transitive If AB ≌ CD and CD ≌ EF,
then AB ≌ EF
How to write a Proof
Proofs are formal statements with a conclusion
based on given information.
One type of proof is a two column proof.
One column with statements numbered;
the other column reasons that are numbered.
Given: EF = GH
Prove EG ≌ FH
#1. EF = GH
E
F
G
#1. Given
H
Given: EF = GH
Prove EG ≌ FH
#1. EF = GH
#2. FG = FG
E
F
G
H
#1. Given
#2. Reflexive Prop.
Given: EF = GH
Prove EG ≌ FH
E
#1. EF = GH
#2. FG = FG
#3. EF + FG = GH + FG
F
G
H
#1. Given
#2. Reflexive Prop.
#3. Add. Prop.
Given: EF = GH
Prove EG ≌ FH
#1.
#2.
#3.
#4.
E
EF = GH
FG = FG
EF + FG = GH + FG
EG = EF + FG
FH = FG + GH
F
G
H
#1. Given
#2. Reflexive Prop.
#3. Add. Prop.
#4.
Given: EF = GH
Prove EG ≌ FH
#1.
#2.
#3.
#4.
E
EF = GH
FG = FG
EF + FG = GH + FG
EG = EF + FG
FH = FG + GH
F
#1.
#2.
#3.
#4.
G
H
Given
Reflexive Prop.
Add. Prop.
Segment Add.
Given: EF = GH
Prove EG ≌ FH
#1.
#2.
#3.
#4.
E
EF = GH
FG = FG
EF + FG = GH + FG
EG = EF + FG
FH = FG + GH
#5. EG = FH
F
#1.
#2.
#3.
#4.
G
H
Given
Reflexive Prop.
Add. Prop.
Segment Add.
#5. Subst. Prop.
Given: EF = GH
Prove EG ≌ FH
#1.
#2.
#3.
#4.
E
EF = GH
FG = FG
EF + FG = GH + FG
EG = EF + FG
FH = FG + GH
#5. EG = FH
#6. EG ≌ FH
F
#1.
#2.
#3.
#4.
G
H
Given
Reflexive Prop.
Add. Prop.
Segment Add.
#5. Subst. Prop.
#6. Def. of ≌
Given: RT ≌ WY; ST = WX
Prove: RS ≌ XY
W
#1. RT ≌ WY
R
X
#1. Given
S
Y
T
Given: RT ≌ WY; ST = WX
Prove: RS ≌ XY
#1. RT ≌ WY
#2. RT = WY
R
W
#1. Given
#2. Def. of ≌
S
X
T
Y
Given: RT ≌ WY; ST = WX
Prove: RS ≌ XY
#1. RT ≌ WY
#2. RT = WY
#3. RT = RS + ST
WY = WX + XY
R
W
S
X
#1. Given
#2. Def. of ≌
#3. Segment Add.
T
Y
Given: RT ≌ WY; ST = WX
Prove: RS ≌ XY
#1. RT ≌ WY
#1.
#2. RT = WY
#2.
#3. RT = RS + ST
#3.
WY = WX + XY
#4. RS + ST = WX + XY
R
W
S
X
T
Y
Given
Def. of ≌
Segment Add.
#4. Subst. Prop.
Given: RT ≌ WY; ST = WX
Prove: RS ≌ XY
#1. RT ≌ WY
#1.
#2. RT = WY
#2.
#3. RT = RS + ST
#3.
WY = WX + XY
#4. RS + ST = WX + XY
#5. ST = WX
#5.
R
W
S
X
T
Y
Given
Def. of ≌
Segment Add.
#4. Subst. Prop.
Given
Given: RT ≌ WY; ST = WX
Prove: RS ≌ XY
#1. RT ≌ WY
#1.
#2. RT = WY
#2.
#3. RT = RS + ST
#3.
WY = WX + XY
#4. RS + ST = WX + XY
#5. ST = WX
#5.
#6. RS = XY
#6.
R
W
S
X
T
Y
Given
Def. of ≌
Segment Add.
#4. Subst. Prop.
Given
Subtract. Prop.
Given: RT ≌ WY; ST = WX
Prove: RS ≌ XY
#1. RT ≌ WY
#1.
#2. RT = WY
#2.
#3. RT = RS + ST
#3.
WY = WX + XY
#4. RS + ST = WX + XY
#5. ST = WX
#5.
#6. RS = XY
#6.
#7. RS ≌ XY
#7.
R
W
S
X
T
Y
Given
Def. of ≌
Segment Add.
#4. Subst. Prop.
Given
Subtract. Prop.
Def. of ≌
Given:
Prove:
x is the midpoint of MN and MX = RX
XN = RX
#1. x is the midpoint of MN
#1. Given
Given:
Prove:
x is the midpoint of MN and MX = RX
XN = RX
#1. x is the midpoint of MN
#1. Given
#2. XN = MX
#2. Def. of midpoint
Given:
Prove:
x is the midpoint of MN and MX = RX
XN = RX
#1. x is the midpoint of MN
#1. Given
#2. XN = MX
#2. Def. of midpoint
#3. MX = RX
#3. Given
Given:
Prove:
#1.
#2.
#3.
#4.
x is the midpoint of MN and MX = RX
XN = RX
x is the midpoint of MN
XN = MX
#2.
MX = RX
#3.
XN = RX
#4.
#1. Given
Def. of midpoint
Given
Transitive Prop.
Something with Numbers
If AB = BC and BC = CD, then find BC
A
D
3X – 1
2X + 3
B
C
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