do now: here we see that point y is between x and z: what does the segment addition postulate tell...

Do Now:

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

What does the Segment Addition Postulate tell us about X, Y, and Z?

YX

Z

XY + YZ = XZ

Segment Addition Postulate

If B is between A and C,

then AB + BC = AC

A B CB

A B C

Remember: Each is a measurement of distance!

Segment Subtraction Postulate

If B is between A and C, thenAB = AC - BC and BC = AC - AB

A B C A B C

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.

Substitution Postulate

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

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.

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How do we use postulates to show congruence?

Given:

B is between A and C

AB=5

AC=10

Prove: AB BC

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Statements Reasons1. AB=5 1. GivenStatements Reasons1. AB=5 1. Given2. AC=10 2. Given

Statements Reasons1. AB=5 1. Given2. AC=10 2. Given3. BC = AC - AB 3. Segment

Subtraction Postulate

Statements Reasons1. AB=5 1. Given2. AC=10 2. Given3. BC = AC - AB 3. Segment

Subtraction Postulate

4. BC = 10 - 5 4. Substitution Postulate

Statements Reasons1. AB=5 1. Given2. AC=10 2. Given3. BC = AC - AB 3. Segment

Subtraction Postulate

4. BC = 10 - 5 4. Substitution Postulate

5. BC = 5 5. Subtraction

6. AB BC6. Definition of Congruence

Statements Reasons1. AB=5, AC=10 1. Given2. B btwn A, C 2. Given3. BC = AC - AB 3. Segment

Subtraction Postulate

4. BC = 10 - 5 4. Substitution Postulate

5. BC = 5 5. Subtraction

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Rays and Angles

• Take out yesterday’s sheet!

Angle Addition Postulate

If ray is between ray and ray , then

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AB−−>

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AC−−>

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AD−−>

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m∠CAB + m∠BAD = m∠CAD

These are all measurements!

B

A D

C

Angle Subtraction Postulate

• If ray is between ray and ray , then

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AB−−>

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AC−−>

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AD−−>

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m∠CAD−m∠CAB = m∠BAD

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m∠CAD−m∠BAD = m∠CAB

Do Now:

• Given:– S is between R and T– X is between S and R– TR=50– TS=20– XS=10

• Prove:– XR TS

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What are Mathematical Relations?

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

This is a relation.

Examples:

This is a relation.

Azim is taller than Sara 3 divides 15

Why are “Congruence” and “Equality” so similar?

• They share a lot of the same properties.

• They are both Equivalence Relations

A relation that meets certain requirements

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

The relation must be all three at once!

Let’s look at these properties…

• We’ll use “=“ as an example and show that it IS an equivalence relation!

A Relation is Reflexive when:• An object is related to itself!

Example: x = x

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

Example: If a = b, then b = a

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!

So…

• Since the relation “=:”– Is reflexive…– Is symmetric…– Is transitive…

We can conclude “=“ is an equivalence relation.

Is “Congruence” an equivalence relation?

YOU BET YOUR SWEET BIPPY, IT IS!

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!