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Definitions 
If . . . Then
Statements
If a ray bisects an angle, . . .
then it divides the angle into two
congruent angles.
If a ray divides an angle into two
congruent angles, . . .
then the ray bisects the angle.
If a line ( ray/segment/point)
bisects a segment, . . .
then it divides the segment into
two congruent segments.
If a line (ray/segment/point)
divides a segment into two
congruent segments, . . .
then the line (ray/segment/point)
bisects the segment.
If a point divides a segment into
two congruent segments, . . .
then it is the midpoint of the
segment.
If a point is the midpoint of a
segment, . . .
then it divides the segment into
two congruent segments.
Definitions 
(cont.)
If . . . Then
Statements
If two rays trisect an angle, . . .
then they divides the angle into
three congruent angles.
If two rays divide an angle into
three congruent angles, . . .
then the rays trisect the angle.
2-5 Proving Angles Congruent
Angle Pairs
Vertical Angles
two angles whose
sides form two
pairs of opposite
rays.
Adjacent Angles
two coplanar
angles with a
common side, a
common vertex,
and no common
interior points.
3
1
2
4
1 and 2 are vertical angles.
3 and 4 are vertical angles.
5
6
5 and 6 are adjacent angles.
2-5 Proving Angles Congruent
Angle Pairs
Complementary Angles
two angles whose
measures add to 90.
(Not necessarily
adjacent.) Each is the
complement of the other.
Supplementary Angles
two angles whose
measures add to 180.
(Not necessarily
adjacent.) Each is the
supplement of the
other.
60
A
1
30
B
2
1 and 2 are complement ary angles.
A and B are complement ary angles.
120
D
60
C
5 and 6 are supplement ary angles.
C and D are supplement ary angles.
5
6
2-5 Five Angle Theorems
Vertical Angles Theorem
Vertical angles are
congruent.
1  2
3  4
3
1
2
4
2-5 Five Angle
Congruent Supplements Theorem
If two angles are supplements of
the same angle, then the two
angles are congruent.
Abbreviation in proofs :
supps s 
Theorems
Congruent Supplements Theorem
If two angles are supplements of
congruent angles, then the two angles
are congruent.
Abbreviation in proofs :
supps  s 
B
3
C
A
1
A and 1 are supplementary.
B and 1 are supplementary.
Therefore, A  B.
C and 3 are supplementary.
D and 4 are supplementary.
3  4
Therefore, C  D.
4
D
2-5 Five Angle
Congruent Complements Theorem
If two angles are complements of
the same angle, then the two
angles are congruent.
Abbreviation in proofs :
comps s 
Theorems,
Congruent Complements Theorem
If two angles are complements of
congruent angles, then the two angles
are congruent. Abbreviation in proofs :
comps  s 
1
A
B
A and 1 are complementary.
B and 1 are complementary.
Therefore, A  B.
cont.
C
3
4
D
C and 3 are complementary.
D and 4 are complementary.
3  4
Therefore, C  D.
2-5 Five Angle
Right Angle Theorem
All right angles are congruent.
Abbreviation in proofs :
all rt. s 
Theorems,
cont.
Congruent and Supplementary
Theorem
If two angles are congruent and
supplementary, then each is a
right angle.
Abbreviation in proofs :
s  and supp.  rt.
Proving the Five Theorems
Vertical Angles Theorem
Given : 1 and 2 are vertical angles
Prove : 1  2
3
1
2
Statement
Reason
1) 1 and 2 are vertical s.
1) Given.
2) m3  m3
3) m1  m3  180
m2  m3  180
2) Reflexive POE
4) m1  m3  m2  m3
4) Substituti on POE
5) m1  m2
5) Subtraction POE
6) 1  2
6) Definition of congruence
3) Angle addition postulate
supps s 
2
Congruent Supplements Theorem
(Same Angle)
1
Given : 1 supp 2 AND 3 supp 2
Prove : 1  3
3
Statement
1) 1 supp 2 ;
3 supp 2
Reason
2) m1  m2  180
m3  m2  180
2) Def. supp s
3) m1  m2  m3  m2
3) Substituti on POE
4) m1  m3
4) Subtraction POE
5) 1  3
5) Definition of congruence
1) Given.
supps  s 
1
Congruent Supplements Theorem
(Congruent Angles)
Given : 1 supp 2 ;
3 supp 4;
2  4
Prove : 1  3
2
3
Statement
1) 1 supp 2 ;
3 supp 4;
2  4
2) m2  m4
Reason
3) m1  m2  180
m3  m4  180
3) Def. supp s
4) m1  m2  m3  m4
4) Substituti on POE
5) m1  m2  m3  m2
5) Substituti on POE
6) m1  m3
6) Subtraction POE
7) 1  3
7) Definition of congruence
1) Given.
2) Definition of congruence
4
All rt. s 
All Right Angles Congruent
Given:
Prove:
B
A is a right angle.
B is a right angle.
A  B
A
Statements
1. A is a right angle;
B is a right angle
2. mA = 90; mB = 90
3. mA = mB
4. A z B
Reasons
1. Given
2. Def. rt. 
3. Substitution POE
4. Def. congruence
s  and supp.  rt.
Angles Both Congruent and
Supplementary are Right Angles
Given:
Prove:
Y
X zY ; X supp.Y
X and Y are right s
X
Statements
1. X zY ; X supp.Y
2. mX= mY
3. mX+ mY=180
4. mX+ mX=180
5. 2mX=180
6. mX=90
7. mY=90
8. X is a rt. ; Y is a rt. 
Reasons
1. Given
2. Definition of congruence
3. Def. supp. s
4. Substitution POE
5. Combine like terms (simplify)
6. Division POE
7. Substitution POE
8. Def. rt. 
Using the
Theorems
Now you can use
these five theorems
as part of other
proofs.
In the REASON
column you can now
write the short form
abbreviations.
Complementary/Supplementary Proof
C
Given : Diagram
3  1.
F
Prove : 2 is supplement ary to 1.
2
3
A
G
Statements
E
D
4
J
B
1
H
Reasons
1.
Diagram; 3  1
1.
Given
2.
FJD is a straight angle
2.
Assumed from diagram
3.
2 and  3 are supp.
3.
If two s form a straight ,
then they are supp.
4.
2 and  1 are supp.
4.
Substitution POC
More Practice
Given : Diagram
X
XOB  YOB.
B
A
Prove : AOX is supplement ary to YOB.
O
Y
Statements
Reasons
1.
Diagram; XOB  YOB
1.
Given
2.
AOB is a straight .
2.
Assumed from diagram.
3.
AOX and XOB are supp.
3.
If two s form a straight,
then they are supp.
4.
AOX and  YOB are supp.
4.
Substitution POC
Using the Theorems
X
Y
Given : Diagram; AX  AB;
BY  AB; 1  2.
Prove : 3  4.
Statements
1. Diagram.
2. AX  AB.
3
1
2
A
3. XAB is a rt. .
Reasons
1. Given.
2. Given.
3. If a ray and line are ,
4. 3 and 1 are compl.
then they form a rt. .
4. If 2 s form a rt. ,
5. BY  AB.
6. YBAis a rt. .
7. 2 and 4 are compl.
then they are comp.
5. Given.
6. If a ray and line are ,
then they form a rt. .
7. If 2 s form a rt. ,
then they are comp.
8. 1  2.
8. Given.
9. 3  4.
9. Comp.  s are  .
4
B