Benchmark 11
I can prove statements
about segments and
angles
Theorem
• A true statement that follows as a
result of other true statements.
• Theorems must be ______
proven in order to
use them.
• Once theorems are proven they
become properties or ________
short cuts that
can be used to prove future
theorems.
Theorem: 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.
Theorem-Segment congruence is symmetric
Given: PQ  XY
Prove: XY  PQ
X
P
Q
Statements
1. PQ  XY
2. PQ = XY
3. XY = PQ
4. XY  PQ
Y
Reasons
1. Given
2. Def. of  Segments
3. Symmetric
4. Def. of  Segments
Given: LK=5, JK=5, JK  JL
Prove: LK  JL
J
5
K
5
L
Statements
Reasons
1. LK=5, JK=5, JK  JL 1. Given
2. Transitive
2. LK = JK
3. LK  JK
3. Def. of  Segments
4. LK  JL
4. Transitive
Given: RT  WY,
ST = WX
S
Prove: RS  XY T
Statements
R
X
Y
W
Reasons
1. RT  WY, ST = WX
1. Given
2. RT = WY
3. RT = RS+ST
WY = WX+XY
4. RS+ST = WX+XY
5. RS = XY
2. Def. of  Segments
3. Seg. Add Post.
4. Substitution
5. Subtraction
6. RS  XY
6. Def. of  Segments
Solve for the variable using the given
information. Given: AB  BC, CD  BC
A 2x+1 B
C 4x-11 D
Statements
Reasons
1. AB  BC, CD  BC
2. AB = BC, CD = BC
3. AB = CD
4. 2x+1 = 4x-11
5. 1 = 2x-11
6. 12 = 2x
7. 6 = x
1. Given
2. Def of  Segments
3. Transitive
4. Substitution
5. Subtraction
6. Addition
7. Division
CONGRUENCE OF ANGLES
THEOREM
THEOREM 2.2 Properties of Angle Congruence
Angle congruence is r ef lex ive, sy mme tric, and transitive.
Here are some examples.
REFLEX IVE
For any angle A,
SYMMETRIC If
A 
B, then
TRANSITIVE If
A 
B and
B 
C, then
A 
A
B 
A
A 
C
Using the Transitive Property
This two-column proof uses the Transitive Property.
GIVEN
m
3 = 40°,
PROVE
m
1 = 40°
1
2,
Statements
1
2
m
1
3
3
m
1=m
4
m
1 = 40°
3
Reasons
3 = 40°,
2
3
1
2
2,
Given
Transitive property of Congruence
3
Definition of congruent angles
Substitution property of equality
Proving Theorem 2.3
THEOREM
THEOREM 2.3 Right Angle Congruence Theorem
All right angles are congruent.
You can prove Theorem 2.3 as shown.
GIVEN
1 and
PROVE
1
2 are right angles
2
Proving Theorem 2.3
GIVEN
1 and
PROVE
1
2 are right angles
2
Statements
1 and
1
Reasons
2 are right angles
2
m
1 = 90°, m
3
m
1=m
4
1
2
2
2 = 90°
Given
Definition of right angles
Transitive property of equality
Definition of congruent angles