September 29, 2010
2.6 Prove Statements about Segments and Angles
Proof : a logical argument that shows a statement is true.
Two–Column Proof: has numbered statements and
corresponding reasons that sow an argument in
a logical order.
Theorem : A true statement that follows from other true statements
*(must be proven)*
Thm 2.1 Congruence of Segments
Segment congruence is reflexive, symmetric, and transitive.
Reflexive : AB ≅ AB
Symmetric : If AB ≅ CD then CD ≅ AB
Transitive : If AB ≅ CD and CD ≅ EF then AB ≅ EF
Ex 1
Given : EF = GH
Prove : EG ≅ FH
E
•
F
•
G
Statements
1. EF = GH
Reasons
1. Given
2. EF + FG = EG
2. Segment Addition Postulate
3. FG + GH = FH
3. Segment Addition Postulate
4. GH + FG = EG
5. EG = FH
4. Substitution Property of Equality
(using statement 1 and 2)
5. Subst. Prop. of Eq
6. EG ≅ FH
6. Definition of Congruent Segments
H
September 29, 2010
Ex 2
Complete the proof.
Given : RT ≅ WY, ST = WX
Prove : RS ≅ XY
Statements
•
S
R
W
Reasons
•
X
T
Y
1. RT ≅ WY
1. Given
2. RT = WY
2. Definition of congruent (≅) segments
3. RT = RS + ST
WY = WX +XY
3. Segment Addition Postulate
4. RS + ST = WX + XY
4. Subst. Prop. of Eq.
5. ST = WX
5. Given
6. RS = XY
6. Subtraction Prop. of Eq.
7. RS ≅ XY
7. Definition of ≅ segments
Ex 3
Given : X is the midpoint of MN and MX = RX
Prove : XN = RX
S
M
X
R
N
STATEMENTS
REASONS
1. X is the midpoint of MN
1. Given
2. MX = RX
2. Given
3. MX = XN
3. Definition of midpoint of a segment.
4. XN = RX
4. Transitive Prop. of Eq. from
statements 2 and 3
(or Subst. Prob. of Eq.)
Homework: Pages 116 – 117 # 1 – 12
September 29, 2010
2.6 Prove Statements about Segments and Angles (continued)
Thm 2.2 Congruence of Angles
Angle congruence is reflexive, symmetric, and transitive.
Reflexive : For any angle A, ∠A ≅ ∠A.
Symmetric : If ∠A ≅ ∠B, then ∠B ≅ ∠A
Transitive : If ∠A ≅ ∠B, and ∠B ≅ ∠C, then ∠A ≅ ∠C
Ex 1: Given: HI = 9, IJ = 9, IJ ≅ JH
Prove: HI ≅ JH
Statements
Reasons
1. HI = 9
1. Given
IJ = 9
IJ ≅ JH
2. HI = IJ
2. Transitive POE
3. HI ≅ IJ
Definition of Congruent
3. Segments
4. HI ≅ JH
4. Transitive POC
(Property of Congruence)
Ex 2: Given: ∠3 and ∠2 are complementary.
m∠1 and m∠2 = 90º
Prove: ∠1 ≅ ∠3
Statements
∠3 and ∠2 are
1. complementary.
1. Given
2. m∠1 and m∠2 = 90º
2. Given
3.
3. Definition of
complementary ∠s.
m∠3 and m∠2 = 90º
Reasons
4. m∠1 + m∠2 = m∠3 + m∠2 4. Transitive Prop. of Equality
5.
m∠1 = m∠3
6.
∠1 = ∠3
5. Subtraction POE
6. Definition of ≅ ∠s