1. Definitions, Postulates, and Theorems used in PROOFS

2. Definition of Congruence
Two figures are congruent IF AND ONLY IF
their measures are equal. (Def ) C D A B
AB = CD
Given or other
AB  CD
Given or other
AB = CD
Def 
Def 
AB  CD OR

3. Definition of Midpoint
If M is the midpoint of AB,
then AM = MB A B M J K F
M is the midpoint of AB
Given
AM = MB
Definition of Midpoint
JF = FK
Given (or other)
Definition of Midpoint
F is the midpoint of JK

4. Definition of Segment Bisector
A point, line, ray, segment or plane that intersects a segment at its midpoint. A X B C
Given
CX bisects AB
X is the midpoint of AB
Definition of Segment Bisector
AX = BX
Definition of Midpoint

5. Definition of an Acute Angle
An angle is an acute angle
if and only if
It measures between 0° and 90°.
GHI is an acute angle.
Given (or other)
0° < mGHI < 90°
Definition of Acute Angle
m7 = 65°
Given (or other)
7 is an acute angle.
Definition of Acute Angle

6. Definition of Right Angle
An angle is a right angle
if and only if
It measures 90°.
ABC is a right angle.
Given (or other)
m  ABC = 90°
Definition of Right Angle
m  7 = 90°
Given (or other)
 7 is a right angle.
Definition of Right Angle

7. Definition of an Obtuse Angle
An angle is an obtuse angle
if and only if
It measures between 90° and 180°.
JKL is an obtuse angle.
Given (or other)
90° < mJKL < 180°
Definition of Obtuse Angle
m  8 = 165°
Given (or other)
 8 is an obtuse angle.
Definition of Obtuse Angle

8. Definition of Complementary Angles
Two angles are complementary
if and only if
the sum of their measures is 90°.
1.  1 and  3 are
complementary angles
1. Given or other
2.Definition of
Complementary Angles
2. m1 + m3 = 90°
1. m 1 + m3 = 90°
1. Given or other
2. 1 and 3 are
complementary angles
2.Definition of
Complementary Angles

9. Definition of Supplementary Angles
Two angles are supplementary
if and only if
the sum of their measures is 180°.
1.  1 and  3 are
supplementary angles
1. Given or other
2.Definition of
Supplementary Angles
2. m1 + m3 = 180°
1. m1 + m3 = 180°
1. Given or other
2.  1 and  3 are
supplementary angles
2.Definition of
supplementary Angles

10. Definition of Linear Pair
If two angles are adjacent and form a line, then they are a linear pair. A B M D
AMD & DMB are a linear pair
Definition of linear pair (Def. LP)

11. Definition of Vertical Angles
Two nonadjacent angles formed by intersecting lines are vertical angles. A B M D C
AMC & DMB are vertical angles
Definition of vertical angles
(Def. Vert. s)

12. Definition of Perpendicular Lines
Two lines are perpendicular if and only if they form a right angle. m n 1 2 3 4
m  n
Given or other…
1 is a right angle
Def  lines
1 is a right angle
Given or other…
m  n
Def  lines

13. Segment Addition Postulate
If T is between V and W,
then VT + TW = VW. W T V
Segment Addition
Postulate
VT + TW = VW

14. Angle Addition Postulate
If X is in the interior of JKL, then mJKX + mXKL = mJKL. X J K L
Angle Addition
Postulate
mJKX + mXKL = mJKL

15. Linear Pair Postulate
If two angles form a linear pair, then they are supplementary. A B M D
AMD & DMB are a linear pair
Definition of linear pair (Def. LP)
AMD & DMB are supplementary.
Linear Pair Postulate
(LP Post.)

16. Corresponding Angles Postulate
If two parallel lines are are cut by a transversal,
then each pair of corresponding angles are congruent.
(Corr. s Post.) g j 1 2 3 4 5 6 h 7 8
1. j ll h
1. Given or other
2. 1   5
2. Corresponding Angles
Postulate (Corr. s Post.)
2   6
3   7
4   8

17. Corresponding Angles Converse Postulate
If two lines are cut by a transversal so that ,
corresponding angles are congruent, then the lines are parallel. (Corr. s Conv. Post.) g j 1 2 3 4 5 6 h 7 8
1. Given or other
1. 1   5
2. j ll h
2. Corresponding Angles
Converse Postulate
(Corr. s Conv. Post.)

18. Right Angle CongruenceTheorem
All right angles are congruent.
1 is a right angle
2 is a right angle
Given or other…
1  2
Right Angle  Th
(Rt   Th)

19. Congruent Supplements Theorem
If two angles are supplementary to the same angle (or to congruent angles) then they are congruent.
1 & 3 are supp.
2 & 3 are supp.
Given or other…
Congruent Supplements
Theorem
1  2 OR
( Supp.Th)

20. Congruent SupplementsTheorems
If two angles are supplementary to the same angle (or to congruent angles) then they are congruent.
1 & 3 are supp.
2 & 4 are supp.
Given or other…
Given or other…
1  2
Congruent Supplements
Theorem
3  4
( Supp.Th)

21. Congruent Complements Theorem
If two angles are complementary to the same angle (or to congruent angles) then they are congruent.
1 & 3 are comp.
2 & 3 are comp.
Given or other…
Congruent Complements
Theorem
1  2 OR
( Comp.Th)

22. Congruent Complements Theorem
If two angles are complementary to the same angle (or to congruent angles) then they are congruent.
1 & 3 are comp.
2 & 4 are comp.
Given or other…
1  2
Given or other…
Congruent Complements
Theorem
3  4
( Comp.Th)

23. Vertical Angles Theorem
Vertical Angles are congruent A B M D C
Definition of vertical angles (Def. Vert. s)
AMC & DMB are vertical angles
Vertical Angles Theorem (Vert. s Th)
AMC  DMB

24. Theorem about Perpendiculars
If two lines intersect to form a linear pair of congruent angles, then the lines are perpendicular.
(2 lines that form  adj linear pr s are .) g 1 2 h
Given or other
1 & 2 are a LP
Given or other
1  2
TH:2 lines that form  adj
linear pr s are .
g  h

25. Theorem about Complements
If two sides of two adjacent acute angles are perpendicular, then the angles are complementary.
(If 2 sides of adj acute s are  then the s are comp.) g 1 2 h
g  h
Given or other
1 & 2 are Comp.
TH: If 2 sides of adj acute s
are  then the s are comp.

26. Theorem about Perpendiculars
If two lines are perpendicular,
then they intersect to form 4 right angles.
( lines form 4 rt. s.) g 1 2 h 4 3
1. g  h
1. Given or other
2. 2 is a right angle
2. Def.  lines
3. 1 is a right angle
3. TH:  lines form 4 rt. s.
3 is a right angle
4 is a right angle

27. Alternate Interior Angles Theorem
If two parallel lines are cut by a transversal,
then each pair of alternate interior angles are congruent.
(Alt. Int. s Th.) g j 1 2 3 4 5 6 h 7 8
1. j ll h
1. Given or other
2. 3   6
2. Alternate Interior Angles
Theorem (Alt. Int. s Th.)
4   5

28. Alternate Exterior Angles Theorem
If two parallel lines are cut by a transversal,
then each pair of alternate exterior angles are congruent.
(Alt. Ext. s Th.) g j 1 2 3 4 5 6 h 7 8
1. j ll h
1. Given or other
2. 1   8
2. Alternate Exterior Angles
Theorem (Alt. Ext. s Th.)
2   7

29. Consecutive Interior Angles Theorem
If two parallel lines are cut by a transversal,
then each pair of consecutive interior angles are supplementary.
(Cons. Int. s Th.) g j 1 2 3 4 5 6 h 7 8
1. j ll h
1. Given or other
2. 3 and 5 are
supplementary
2. Consecutive Interior
Angles Theorem
(Cons. Int. s Th.)
4 and 6 are
supplementary

30. Perpendicular Transversal Theorem
If a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other.
( Transv. Th) g j h
1. g  h
1. Given or other
2. j ll h
2. Given or other
3. g  h
Perpendicular Transversal
Theorem ( Transv. Th.)

31. Alternate Interior Angles Converse Theorem
If two lines are cut by a transversal so that
alternate interior angles are congruent, then the lines are parallel. (Alt. Int. s Conv. Th.) g j 1 2 3 4 5 6 h 7 8
1. 3   6
1. Given or other
2. j ll h
2. Alternate Interior Angles
Converse Theorem
(Alt. Int. s Conv. Th.)

32. Alternate Exterior Angles Converse Theorem
If two lines are cut by a transversal so that
each pair of alternate exterior angles are congruent, then the lines are parallel.
(Alt. Ext. s Conv. Th.) g j 1 2 3 4 5 6 h 7 8
1. Given or other
1. 1   8
2. j ll h
2. Alternate Exterior Angles
Converse Theorem
(Alt. Ext. s Conv. Th.)

33. Consecutive Interior Angles Converse Theorem
If two lines are cut by a transversal so that
consecutive interior angles are supplementary, then the lines are parallel.
(Cons. Int. s Conv. Th.) g j 1 2 3 4 5 6 h 7 8
1. 3 and 5 are
supplementary
1. Given or other
2. j ll h
2. Consecutive Interior
Angles Converse Theorem
(Cons. Int. s Conv. Th.)

34. A Parallel Theorem 1. g ll h 1. Given or other 2. j ll h
If two lines are parallel to the same line, then they are parallel to each other.
(2 lines ll to same line are ll) g j h
1. g ll h
1. Given or other
2. j ll h
2. Given or other
3. j ll g
3. Two lines parallel to same
line are parallel
(2 lines ll to same line are ll)

35. Another Parallel Theorem
If two lines are perpendicular to the same line, then they are parallel to each other.
(2 lines  to same line are ll) g j h
1. g  j
1. Given or other
2. Given or other
2. g  h
3. j ll h
3. Two lines perpendicular to
same line are parallel
(2 lines  to same line are ll)

36. 6 Ways to Prove lines Parallel

37. Segment Proof Prove: AC  BC Prove: AC  BC 1. GivenD E
Given: AE  BD, CD  CE
AE  BD
CD  CE
Prove: AC  BC
Prove: AC  BC
1. Given
1. AE  BD, CD  CE
2. Def. 
2. AE =BD, CD = CE
AE = AC + CE
BD = BC + CD
3. Segment Add. Post =
AC + CE
BC + CD
4. Substitution Prop.=
AC = BC
5. Subtraction Prop.=
AC  BC
6. Def. 