Do Now:

1
Do Now

• Here we see that point Y is between X and Z

X
Y
Z
What does the Segment Addition Postulate tell us
about X, Y, and Z?
XY YZ XZ
2
Segment Addition Postulate

• If B is between A and C,
• then AB BC AC

3
Segment Subtraction Postulate

• If B is between A and C, then
• AB AC - BC and BC AC - AB

A
B
C
A
B
C
4
What IS a postulate?

• Definition A postulate is a statement that we
  accept without proof.

But then, what do we call something that we need
to prove?
Definition A theorem is a statement that must be
proven before we can accept it.
5
Substitution Postulate

• A quantity may be substituted for its equal in
  any expression.

6
Congruence What is it?

• Two objects are congruent ( )if their
  measurements are equal.
• Later on, two objects will be congruent if each
  of their parts has the same measurement.

What is the difference between two things being
equal and being congruent?
Congruent means two things are EXACT
copies. Equal means they are the SAME THING.
7
How do we use postulates to show congruence?
Given B is between A and C AB5 AC10
Prove AB BC
8
Rays and Angles

• Take out yesterdays sheet!

9
Angle Addition Postulate

• If ray is between ray and ray ,
  then

10
Angle Subtraction Postulate

• If ray is between ray and ray ,
  then

11
Do Now

• Given
• S is between R and T
• X is between S and R
• TR50
• TS20
• XS10
• Prove
• XR TS

12
What are Mathematical Relations?

• Definition An association or comparison between
  two objects (like numbers or shapes).

13
Examples
Azim is taller than Sara
3 divides 15
14
Why are Congruence and Equality so similar?

• They share a lot of the same properties.
• They are both Equivalence Relations

15
What is an equivalence relation?

• A relation R (on some set of mathematical
  objects) is an equivalence relation if
• R is reflexive
• R is symmetric
• R is transitive

16
Lets look at these properties

• Well use as an example and show that it IS
  an equivalence relation!

17
A Relation is Reflexive when

• An object is related to itself!

Example x x
18

• A Relation is Symmetric when
• A relation can be expressed in either order.

Example If a b, then b a
19
A Relation is Transitive if

• A is related to B, and B is related to C, then A
  is related to C.

Example If x y and y z, then x z.
This is like
The Law of Syllogism!
20
So

• Since the relation
• Is reflexive
• Is symmetric
• Is transitive

We can conclude is an equivalence relation.
21
Is Congruence an equivalence relation?

• YOU BET YOUR SWEET BIPPY, IT IS!

22
Homework

• Pg. 123-124
• 1-8, 10, 12, 14, 16
• Show examples to support your assertions!
• So if something is NOT reflexive, show show an
  example!