Coordinate Algebra—Proof Properties
Prior Knowledge
The first statement which you know
because it’s given to you
When you do very simple math like
combining like terms
Angles are congruent if and only if they
are equal.
If mA  mB , then A  B
Segments are congruent if and only if
they are equal.
If AB  CD , then AB  CD
If A  B , then A  C  B  C
1. Given
2. Simplify
3. Definition of congruent angles.
4. Definition of congruent segments
5. Addition Property of Equality
If A  B , then A  C  B  C
6. Subtraction Property of Equality
If A  B , then AC  BC
7. Multiplication Property of Equality
A B

C C
If AB  C  , then AB  AC .
8. Division Property of Equality
If A  B , then A may be substituted for
B anywhere that it appears.
If a point is the midpoint of a segment,
then it divides the segment into to 
segments.
If an angle is a right angle, then it’s
measure is 90o.
If angles are complementary, then their
sum is 90o.
If angles are supplementary, then their
sum is 180o.
10. Substitution Property of Equality
Anything is equal or congruent to itself.
A  A or A  A
15. Reflexive Property of
Congruence / Equality
If A  B , then B  A .
If A  B , then B  A .
16. Symmetric Property of
Congruence / Equality
If A  B and B  C , then A  C .
If A  B and B  C , then A  C .
17. Transitive Property of
Congruence / Equality
If B is between A and C, then
AB  BC  AC .
If B is in the interior of ADC ,
then ADB  BDC  ADC .
18. Segment Addition Postulate
If two angles are supplementary to the
same or  angles, they are  .
20. Congruent Supplements
Theorem
If A  B , then
9. Distributive Prop. of Equality
11. Definition of a Midpoint
12. Definition of a right angle
13. Definition of complementary
angles
14. Definition of supplementary
angles
19. Angle Addition Postulate
If two angles are complementary to the
same or  angles, they are  .
All right angles are congruent.
21. Congruent Complements
Theorem
22. Right Angles Congruence Thm.
If two angles form a linear pair, then
they are supplementary.
23. Linear Pair Postulate
If two lines are perpendicular, then they
intersect to form four right angles.
Vertical angles are congruent.
24. Definition of perpendicular lines
If a transversal is  to one of two ||
lines, it is  to the other.
26. Perpendicular Transversal
Theorem
If two lines are  to the same line then
they are ||.
27. Lines Perpendicular to a
Transversal Theorem
If two || lines are cut by a transversal,
then alternate interior angles are  .
28. Alternate Interior Angles Theorem
If two || lines are cut by a transversal,
then corresponding angles are  .
29. Corresponding Angles Postulate
The sum of the interior angles of a
triangle is 180o.
If two triangles are congruent, then all
their corresponding parts are congruent
as well.
If all three sides of two triangles are  ,
then the triangles are  .
30. Triangle Sum Theorem
If two sides and the included angle of
two  ’s are  , then the  ’s are  .
33. Side-Angle-Side (SAS)
Congruence Postulate
If the hypotenuse one leg of right  ’s
are  , then the  ’s are  .
34. Hypotenuse-Leg (HL)
Congruence Theorem
If two angles and the included side of 
’s are  , then the  ’s are 
35. Angle-Side-Angle (ASA)
Congruence Postulate
If two angles and the nonincluded side
of  ’s are  , then the  ’s are 
36. Angle-Angle-Side (AAS)
Congruence Theorem
25. Vertical Angle Theorem
31. Corresponding Parts of
Congruent Triangles are Congruent
(CPCTC)
32. Side-Side-Side (SSS)
Congruence Postulate