Download Reason Sheet Chapter 3

Reason Sheet Chapter 3
1. Given
2. Segment Addition Postulate: If B is between A and C, then AB + BC = AC.
3. Angle Addition Postulate: If point B lies in the interior of AOC , then
mAOB  mBOC  mAOC .
4. A line contains at least two points; a plane contains at least three points not all in one line; space
contains at least four points not all in one line.
5. Through any two points there is exactly one line.
6. Through any three points there is at least one plane, and through any three noncollinear points
there is exactly one plane.
7. If two points are in a plane, then the line that contains the points is in that plane.
8. If two planes intersect, then their intersection is a line.
9. If two lines intersect, then they intersect in exactly one point.
10. Through a line and a point not on the line there is exactly one plane.
11. If two lines intersect, then exactly one plane contains the lines.
12. Addition Property of Equality: If a=b and c=d, then a+c=b+d.
13. Subtraction Property of Equality: If a=b and c=d, then a-c=b-d.
14. Multiplication Property of Equality: If a=b, then ca=cb.
15. Division Property of Equality: If a=b and c  0 , then
a b
 .
c c
16. Substitution Property of Equality: If a=b, then either a or b may be substituted for the other in
any equation or inequality.
17. Distributive Property (Equality): a(b+c)=ab+ac
18. Reflexive Property of Equality (Congruence) : a=a ( AB  AB )
19. Symmetric Property of Equality (Congruence): If a=b, then b=a. (If AB  CD , then CD  AB .)
20. Transitive Property of Equality (Congruence): If a=b and b=c, then a=c. (If AB  CD and
CD  EF , then AB  EF .)
21. Definition of Midpoint – Let E be between A and B. E is the midpoint of AB if and only if
AE=EB or AE  EB .
22. Definition of Angle Bisector – OE is an angle bisector of AOC if and only if
AOE  EOC or mAOE  mEOC .
23. Definition of Segment Bisector – If AC bisects BD at point E, then BE=ED.
24. Linear Pair Postulate: If two angles form a linear pair, then they are supplementary.
25. Definition of Supplementary: 1 and 2 are supplementary if and only if
m1  m2  180 .
26. Definition of Complementary: 1 and 2 are complementary if and only if
m1  m2  90 .
27. Definition of Congruent – Angles are congruent if and only if they have the same measure.
Segments are congruent if and only if they have the same length.
28. Vertical Angles Theorem: If 1 and 2 are vertical, then 1  2(m1  m2) .
29. Congruent Supplements Theorem: If two angles are supplements of congruent angles (or the
same angle), then the two angles are congruent.
30. Congruent Complements Theorem: If two angles are complements of congruent angles (or the
same angle), then the two angles are congruent.
31. Definition of Perpendicular – Two lines are perpendicular if and only if the lines intersect to
form right angles (90° angles).
32. All right angles are congruent(equal).
33. If two angles are congruent and supplementary, then each is a right angle.
34. Definition of Right Angle: An angle is right if and only if it has a measure equal to 90 degrees.
35. SSIA Postulate: If two lines cut by a transversal are parallel, then SSIA are supplementary (add
to 180).
36. Corr Thrm: If two lines cut by a transversal are parallel, then corresponding angles are
congruent (=).
37. AIA Thrm: If two lines cut by a transversal are parallel, then alternate interior angles are
congruent (=).
38. AEA Thrm: If two lines cut by a transversal are parallel, then alternate exterior angles are
congruent (=).
39. Converse of Corr: If two lines are cut by a transversal and corresponding angles are congruent
(=), then the lines are parallel.
40. Converse of AIA: If two lines are cut by a transversal and alternate interior angles are congruent
(=), then the lines are parallel.
41. Converse of AEA: If two lines are cut by a transversal and alternate exterior angles are
congruent (=), then the lines are parallel.
42. Converse of SSIA: If two lines are cut by a transversal and same side interior angles are
supplementary (add to 180), then the lines are parallel.
43. If two lines are parallel to the same line, then they are parallel to each other.
44. In a plane, if two lines are perpendicular to the same line, then they are parallel to each other.
45. Perpendicular Transversal Theorem – In a plane, if a line is perpendicular to one of two parallel
lines, then it is also perpendicular to the other.
46. Parallel Postulate: Through a point not on a line there is one and only one line parallel to the
given line.
47. Triangle-Angle Sum Theorem: The sum of the measures of the angles of a triangle is 180.
48. Triangle Exterior Angle Theorem: The measure of each exterior angle of a triangle equals the
sum of the measures of the remote interior angles.
49. Third Angle Theorem: If two angles of one triangle are equal (congruent) to two angles of
another triangle, then the third angles are equal (congruent).