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