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Geometry CP
Lesson 2-5: Postulates and Proofs
Main Ideas:
 Identify and use basic postulates about points, lines, and planes.
 Learning the basic structure of a proof.
CA Standards: 1, 2, 3
In geometry, a postulate or axiom is a statement that is accepted as true. Postulates describe fundamental
relationships in geometry.
Postulate: Through any two points, there is exactly one line.
Postulate: Through any three points not on the same line, there is exactly one plane.
Postulate: A line contains at least two points.
Postulate: A plane contains at least three points not on the same line.
Postulate: If two points lie in a plane, then the line containing those points lies in the plane.
Postulate: If two lines intersect, then their intersection is exactly one point.
Postulate: If two planes intersect, then their intersection is a line.
Ex 1: Some snow crystals are shaped like regular hexagons.
How many lines must be drawn to interconnect all vertices
of a hexagonal snow crystal? __15__
Ex 2: Determine if each statement is always, sometimes, or never true. Explain your answer.
1. If there is a line, then it contains at least 3 noncollinear points.
Never. Points on the same line are collinear._________________________________________
2. If there are points A, B, and C, then there is exactly one plane that contains them.
___Sometimes. The points have to be non-collinear. If they are collinear, there could be two
planes that go through them.___
3. If there are two lines, then they will intersect.
__Sometimes. If lines are parallel, they will not intersect.________________________________
4. If points J and K lie in plane M, then JK lies in M.
Always. Postulate: If two points lie in a plane, then the line containing those points lie in the
plane.
A proof is a logical argument in which each statement you make is supported by a statement that is
accepted as true.
There are different types of proof formats: paragraph, flow-proof, and 2-column. In this class, we will
only use 2-column proofs.
Every 2-column proof has these parts:
 Given (this is the information that is provided to you)
 Prove (this is what you are trying to show is true based on the given information)
 Column of statements (these are the statements that make up your logical argument)
 Column of reasons (these are the supporting statements that “back up” your argument)
Ex 3:
Given: Points H and I
Prove: HI is unique
Statements
1. Points H and I exist
2. HI is unique
Reasons
1. Given
2. Postulate: Through any two points, there is exactly
one line
The reasons in a proof will be definitions, postulates, theorems, and algebraic properties.
Here is a list of some definitions and theorems that you have already learned which will be used
frequently in geometric proofs:
These should be memorized because they frequently come up in writing proofs. Make flash cards
to help you memorize them.
Definition of a right angle: If an angle is a right angle, then its measure is _90o__.
Definition of complementary angles: If two angles are complementary, then their sum is _90o_.
Definition of supplementary angles: If two angles are supplementary, then their sum is _180o.
Supplements Theorem: If two angles form a linear pair, then they are __supplementary____.
Vertical Angles Theorem: If two angles are vertical, then they are ___congruent_______.
Definition of perpendicular lines: If two lines are perpendicular, then they form _right angles.
Definition of Congruency: If QR  ST , then __QR=ST_________
Definition of a Midpoint: If M is the midpoint of AB , then __AM=MB__
Segment Addition Postulate: If K lies between J and L, then _____JK+KL=JL___________
Here’s a new theorem introduced in this lesson to add to the list above.
Midpoint Theorem: If M is the midpoint of AB , then AM  MB .
Below is a proof of the theorem…
Given: M is the midpoint of AB
Prove: AM  MB
Statements
Reasons
1. M is the midpoint of AB
1. Given
2. AM=MB
2. Definition of a midpoint
3. AM  MB
3. Converse of the definition of congruency