Geometry Notes G.3 (2.6, 2.7) Segment, Angle, and Angle Pair Proofs
Mrs. Grieser
Name: ____________________________________________ Date: _____________ Block: _______
Segment and Angle Proofs
Theorems:
 Statements (conjectures) that have been proven.
 Once we have proven a theorem, we can use it in other proofs.
Congruence of Segments Theorems
Segment congruence is reflexive, symmetric, and
transitive.
For any segment AB , AB  AB
Congruence of Angles Theorems
Angle congruence is reflexive, symmetric, and
transitive.

Reflexive: For any angle A, _____________
Symmetric: If AB  CD then CD  AB

Symmetric: If A  B , then ___________
Transitive: If AB  CD and CD  EF then

Transitive: If A  B and B  C then

Reflexive:


AB  EF
Reflexive Theorem of Segment Congruence:
Given: AB is a line segment. Prove: AB  AB
Statement
Reasons
1) Given
1) AB is a line
segment
2) AB  AB
3) AB  AB
2) Reflexive property of =
3) Definition of 
segments
Symmetric Theorem of Segment Congruence:
Given: AB  CD Prove: CD  AB
Statement
Reasons
1 Given
1) AB  CD
___________
Reflexive Theorem of Angle Congruence:
Given: A is an angle. Prove: A  A
Statement
Reasons
1) A is an angle
1) Given
2) mA  mA
2) Reflexive property of =
3) A  A
3) Definition of  angles
Symmetric Theorem of Angle Congruence:
Given: A  B Prove: B  A
Statement
Reasons
1) Given
1) A  B
2) AB  CD
2)______________________
2) mA  mB
2)_______________________
3) CD  AB
3)______________________
3) mB  mA
3)_______________________
4) CD  AB
4)______________________
4) B  A
4)_______________________
Transitive Theorem of Segment Congruence:
Given: AB  CD and CD  EF Prove: AB  EF
Statement
Reasons
1) Given
1) AB  CD
CD  EF
2) AB  CD
CD  EF
2) Definition of  segments
3) AB  EF
3)________________________
4) AB  EF
4)________________________
Transitive Theorem of Angle Congruence:
Given: A  B and B  C Prove: A  C
Statement
Reasons
1) Given
1) A  B
B  C
2) _____________
2) ___________________
_____________
3) ____________
4) A  C
3) ___________________
4) ___________________
Geometry Notes G.3 (2.6, 2.7) Segment, Angle, and Angle Pair Proofs
Mrs. Grieser Page 2
Complete the proofs:
Given: AD = 8, BC = 8;
Given: AD = 12, AB = 12,
BC  CD
BC  CD , AD  CD
Prove: AD  CD
Statement
1) AD = 8
BC = 8
2) AD = BC
Reasons
1) Given
3) AD  BC
3) ________________________
4) BC  CD
4) ________________________
5) AD  CD
5)________________________
2) ________________________
Given: ABC  CBD ,
mCBD  50 ,
Reasons
1) Given
2) ____+_____ = mCBE
2)____________________
mCBE  100
________________
7) ABC  DBE
2) _______________________
3) ________________________
4) _______________________
AD  CD
5) BC  AD
5) _______________________
6) BC  AB
6) _______________________
CD  AD
Prove: Perimeter of ABCD = 4AB
Statement
1) ABC  CBD ,
mCBD  50 ,
50  mDBE  100
mDBE  50
mCBD = ________
3) AD  AB
4) BC  CD
Reasons
1) Given
Given: AB  BC , BC  CD ,
mCBE  100
Prove: ABC  DBE
3)
4)
5)
6)
Prove: BC  AB
Statement
1) AD = 12
AB = 12
2) AD = AB
3) ___________________
4) ____________________
5) __________________
6) __________________
7) ____________________
Statement
1) AB  BC , BC  CD ,
Reasons
1) ____________________
CD  AD
2) AB  BC , BC  CD ,
CD  AD
3) AB  CD , AB  AD
4) Perimeter ABCD =
AB + BC + CD + AD
5) _____________________
6) _____________________
2) ____________________
3) ____________________
4) ____________________
5) __________________
6) __________________
Solve for x. Explain your steps. Given: D  DEG , EG bisects DEF
Angle Pair Proofs

New Postulate:
o Linear Pair Postulate: If two angles form a linear pair, then they are supplementary.
o
1 and 2 form a linear pair, so 1 and 2 are
supplementary and m1  m  2  __________ .
Geometry Notes G.3 (2.6, 2.7) Segment, Angle, and Angle Pair Proofs

Mrs. Grieser Page 3
New Theorems:
Right Angle Congruence Theorem
All right angles are congruent.
Proof: Given: 1 and 2 are right angles
Prove: 1  2
Statements
Reasons
1) __________________

1

2
1)
and
are
right angles
2) m1  90 ;
2) ___________________
m2  90
3) m1  m2
3) ___________________
4) 1  2
4) ___________________
Proof: Given: 1 and 2 are supplements;
3 and 2 are supplements
Prove: 1  3
Statements
Reasons
1) _______________

1

2
1)
and
are
supplements
3 and 2 are
supplements
2)________________
2) m1  m2  180
m3  m2  180
3) m1  m2  m3  m2 3)________________
4)________________
4) m1  m3
5)________________
5) 1  3
Congruent Complements Theorem
If 2 angles are complementary to the same
angle (or to congruent angles), then they are  .
Proof: Given: 1 and 2 are complements
1 and 3 are complements
Prove: 2  3
Statements
Reasons
1)______________
1) 1 and 2 are
complements
1 and 3 are
complements
2)______________
2) m1  m2  90
m1  m3  90
3) m1  m2  m1  m3
4) m2  m3
5) 2  3
Congruent Supplements Theorem
If 2 angles are supplementary to the same angle
(or to congruent angles), then they are congruent.
3)______________
4)______________
5)_____________
Vertical Angles Congruence Theorem
Vertical angles are congruent.
Proof: Given: 1 and 3
are vertical angles
Prove: 1  3
Statements
Reasons
1)_______________
1) 1 and 3 are
vertical angles
2) 1 and 2 are a 2)________________
linear pair;
2 and 3 are a
linear pair
3) Linear pair
3) 1 and 2 are
postulate
supplementary;
2 and 3 are
supplementary
4) Congruent
4) 1  3
supplements theorem
Applications of the Theorems/Postulate
Example 1: Find the angles; what postulates/theorems are used?
a) If m1  112 , find m2 , m3 , and m4
b) If m2  67 , find m1, m3 , and m4
c) If m4  71 , find m1, m2 , and m3
Geometry Notes G.3 (2.6, 2.7) Segment, Angle, and Angle Pair Proofs
Mrs. Grieser Page 4
Example 2: Given: 1 and 2 are complementary;
2 and 3 are complementary;
1 and 4 are complementary
Identify pairs of congruent angles.
Example 3: Find the value of the variables in the figure.
Example 4: Given: S is a right angle;
mRTS  40 , mRTU  50
Prove: S  STU
Statements
1) S is a right angle;
mRTS  40 , mRTU  50
2) _____= mRTS  mRTU
Reasons
1) ________________________________
3) mSTU = _____ + _______
3) Substitution property of =
4) mSTU = _____
5) STU is a right angle.
4) Simplify
5)_________________________________
6) ____________________
6) Right angle congruence theorem
2)_________________________________
Example 5: The figure shows the side view of a picnic table.
Given: 1  4
Prove: 2  3
Statements
1) 1  4
2) 1 and _______ are a linear pair
3 and _______ are a linear pair
3) ________________________
________________________
4) 2  3
Reasons
1) _________________________
2) _________________________
3) Linear pair postulate
4) _________________________
You Try...
a) Find mAEB
b)
Given: 4 is a right angle
Prove: 2 and 4 are supplementary
Statements
1) 4 is a right angle
2) __________________________
3) 2  4
4) __________________________
5) m2  90
6) __________________________
Reasons
1) _________________________
2) Def. of a rt. angle
3) _________________________
4) Def. of  angles
5) _________________________
6) m2  m4  180