2-6 Proving Statements about Segments
Given: AC = BD
Prove: AB = CD.
A
B
C
D
Given: AB = CD
Prove: AC = BD.
A
B
C
D
theorem: proven statement
Linear Pair Theorem: If two angles form a linear pair, then
they are supplementary.
Proof.
Given:
Prove:
D
A
1 and 2 form a linear pair
1 supp 2
Statements
1.
1 and
1.
2.
2.
3.
3.
4.
4.
5.
5.
6.
6.
7.
1 supp
B
Reasons
2 form a linear pair
2
7.
2
1
given
C
A
B
D
C
Congruent Complements Theorem:
If two angles are complementary to the same angle
(or to two congruent angles) then the two angles are
congruent.
Congruent Supplements Theorem:
If two angles are supplementary to the same angle
(or to two congruent angles) then the two angles are
congruent.
(Proof): Congruent Complements Theorem
If 2 angles are complementary to the same angle, then
they are congruent to each other.
Given:
Prove:
Statements
Reasons
(Proof): Congruent Supplements Theorem
If 2 angles are supplementary to the same angle, then
they are congruent to each other.
Given:
Prove:
Statements
Reasons
1. Given: m 1 = 24, m 3 = 24
1 comp. 2
3 comp. 4
Prove:
2 =
Statement
4
Reason
1. __________________ 1. given
2. __________________ 2. given
3. __________________ 3. ____________________
4.
1 = 3
4. ____________________
5. __________________ 5. given
6. __________________ 6. given
7. __________________ 7. _____________________
B
Right Angle Congruence Theorem:
All right angles are congruent.
A
Given: A and B are right angles
Prove: A = B
Statement
1.
A and
Reason
B are right angles 1.
2. m A = 90 ; m B = 90
2.
3. m A = m B
3.
4.
4. Definition of = angles
3. Given:
ABC =
Prove:
DAB and
BCD
DAB =
ABC are rt. angles
D
BCD
A
Statement
Reason
1. _____________________ 1. _____________________
2. _____________________ 2. _____________________
3. _____________________ 3. _____________________
4.
ABC = BCD
C
4. _____________________
5. _____________________ 5. _____________________
B
B
4. Given: 1 = 2,
2= 3
Prove:
3= 4
1= 4
A
Statement
C
Reason
1. ______________________ 1. given
2. ______________________ 2. given
3. ______________________ 3. ______________________
4. 3 = 4
4. given
5. ______________________ 5. _______________________
5. Given: m 1 = 63, 1 = 3
3 = 4
Prove: m 4 = 63
Statement
Reason
1. ___________________ 1. given
2. ___________________ 2. given
3. ___________________ 3. ____________________
4. ___________________ 4. defn. = angles
5. m 1 = 63
5. given
6. ___________________ 6. ________________________
K
6. Given: LK = 5, JK = 5, JK = JL
Prove: LK = JL
J
L
Statement
Reason
1.
1. given
2.
2. given
3. LK = JK
3.
4. LK = JK
4.
5. JK = JL
5.
6.
6.
7. Given: X is the midpoint of MN
MX = RX
S
M
X
Prove: XN = RX
R
Statement
Reason
1.
1.
2.
2. Definition of midpoint
3. MX = XN
3.
4.
4. given
5. XN = RX
5.
N
8. Given: RS = XY
ST = WX
S
R
Prove: RT = WY
W
Statement
Reason
X
1.
1. given
2.
2. given
3.
3. Segment Addition Postulate
4. XY + WX = RT
4.
5. WX + XY = WY
5.
6. RT = WY
6.
T
Y
9. Given: AB = BC, BC = CD
Find BC.
D
A
3x-1
B
2x+3
C
10.
S
R
Given: RT = WY, ST = WX
Prove: RS = XY
W
X
Statement
1. ____________________
Reason
1. given
2. RT = WY
2. ____________________
3. ____________________
3. Segment Addition Postulate
4. ____________________
4. Segment Addition Postulate
5. ____________________
5. _____________________
6. ST = WX
6. given
7. RS = XY
7.
8. RS = XY
8. ______________________
T
Y
11. Given: AD = 8, BC = 8, BC = CD
B
Prove: AD = CD
C
A
Statement
Reason
1. ________________ 1. given
2. BC = 8
2. _______________
3. ________________ 3. transitive
4. ________________ 4. Definition of = segments
5. BC = CD
6. AD = CD
5. ____________________
6. _____________________
D