Transcript GeometryProofs

Geometry Proofs
Math 416
Time Frame
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Definition
Congruent Triangles
Axiom & Proofs
Propositions
Definitions
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Geometric Proofs
The essence of pure mathematics
The creative and artistic center of math
The ability to explain in a detailed concise
logical manner how a proposition
(problem) is either true or false.
Definitions (con’t)
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Detailed – hard facts
Concise – short to the point
Logical – set of rules based on reason
A proof generally falls back to things that
are either known, accepted or already
proven. This is how we attain knowledge
Enlightenment
Gaining Knowledge
Proposition
Proposition
Proposition
Proposition
Definition
Axiom
Thoerem
Definitions
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Definition: You define something once you
identify its essential characteristics
For example, triangle – a two dimensional
polygon with three sides
Not
Must
Axiom
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Axioms: An obvious statement that is
acceptable without proof
For example, the shortest distance
between two points is a straight line
Propositions
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Propositions are statements that require
proof
Once proven they are called theorems
For example Proof
3
1
2
STATEMENT AUTHORITIES
<1 + <3 = 180°
<2 + < 3 = 180°
<1 = <2 = 180
DEFINITION
DEFINITION
ALGEBRA
Theorums
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This proposition now becomes a theorem
Hence, vertically opposite angle theorem
Theorems can be used in a proof as an
authority
Definitions must
use terms that are already defined
Be reversible once you have the
characteristics you have the object
not give unnecessary information
Examples #1 of Definitions
Definition: A belingas is a shape with
a dot on a vertex
are belingas
Which of the following is a belingas?
Example #2 of a Definition
Stencil #1
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Which of the following is a Gatu?
Are Gatus
Definition: A Gatu is a shape with at least
one curved side
Axioms
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A statement not requiring proof
A whole is equal to the sum of its part
Completion
C
A
B
D
< ABD = <ABD + <CBD
• Any quantity can be replaced by
another equal quantity
Axioms
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Easiest thing to do is
to assign numbers to
letters…
a=0;b=4;c=4;q=4
Replacement…
If a + b = c
AND b = q
Then a + q = c
The shortest distance between two points is a
straight line
Only one line can pass through the same two
points
Given a point and a direction, only one line with
that direction can pass through the point
Postulates
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Theorems we will not prove are called
postulates specifically the congruence
postulates
Hypothesis: Given two triangles with
corresponding sides equal we say
CONC: Two triangles are congruent
X
A
ABC
B
CY
Z
YZX
By S S S
Postulates
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Hypothesis: Given two triangles with two
corresponding sides equal and the contained
angle equal
Conclusion: The two triangles are congruent
X
A
ABC
°
By SAS
Y
B
°
C
Z
ZXY
Postulates
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Hypothesis: Given two triangles with
two corresponding angles equal and
the contained side equal
Conclusion: The two triangles are
congruent
A
X
O
B
O
X
CY
X
ABC
Z
ZXY
By ASA
Do #2 
Theorems
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Once again we will not prove
But you may be required to
You should be able to
Theorems
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The 90° completion theorem or the
complementary angle theorem
x
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HYP: Diagram
y
CONC < X + <Y = 90°
The 180° Completion Theorem
HYP Diagram
x
y CONC <x + <y = 180
Vertically Opposite Angle Theorem
1
4
3
2
Conclusion < 1 = < 2
< 3 = <4
Triangle Sum Theorem
1
2
3
Conclusion
<1 + <2 + <3 = 180°
Isosceles Triangle Theorem
1
Conclusion
<1 = <2
2
Given an
isosceles triangle,
the angles
opposite the
equal sides are
equal
Isosceles Triangle Theorem
Converse
A
B
Conclusion
AB = AC
Given an isosceles
triangle, the sides
opposite the equal
angles are equal
C
Note: The
converse is
true also to
prove // lines
Parallel Line Theorem
1
Sometimes called
Corresponding
angles
3
a
c
Conclusion
<4 = < a
<3<b
2
4
b
d <1 < a
<2 = <b
<3 = < c
<4 = <d
<3 + <a = 180°
<4 + <b = 180°
Parallelogram Theorem and
Converse
D
A
x
B
C
In a parallelogram opposite sides
are equal, opposite angles are
equal and the diagonals bisect each
other
Conclusion:
AD = BC Opposite
AB = DC Sides
< BAD = <DCB Opposite
Angles
< ABC = < ADC
BX = XD Diagonals
AX = XC Bisected
Triangle Parallel Similarity Theorem
A
B
C
Conc
ABC ˜
D
E
ADE
Do #3 
Test Question
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If
ABC ˜
XYZ and then < XYZ is
50°, how much is angle ABC?
50°
Vertically opposite angles is an example of
a
a) Theorum b) axiom c) definition
d) postulate
Pythagoras Theorem
A
b
c
Given a right angle triangle,
the square of the hypotenuse
is equal to the sum of the
squares of the other two sides
HYP: Diagram
B
a
C
CONC: b2 = a2 + c2
Pythagoras Examples
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Solve for x
x2
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=
62
+
82
x x2 = 36 + 64
6
x = 10
8
Solve for x
202 = x2 + x2
400 = 2x2
200 = x2
14.14 = x2
x
=
14.14
20
x
x
The 30-60-90 Theorem
A
60°
c
B
The side opposite the 30°
angle is half the hypotenuse.
b
HYP: Diagram
30° C CONC: c = ½ b
OR
b = 2c
The 30-60-90 Theorem Converse
A
b
B
2b
If the hypotenuse is twice
the length of one of the legs,
the angle opposite the leg is
30°
C
HYP: Diagram
CONC: <ACB = 30°
30-60-90 Examples
(2x)2=x2+196
6
4x2=x2+196
Opposite the 30°
30°
It is half the
x hypotenuse
14
x = 12
30°
3x2=196
x2= 65.33
x = 8.08
x
Exam Question
D
A
Hyp: Diagram
Conc: < ABC = < ADC
B
C
Construction AC
Exam Questions Con’t
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Fill in the missing authorities
Statement Authorities
< DAC = <ACB
// Line Theorum
< DCA = <BAC
// Line Theorum
AC = AC
Reflex
Thus
DAC
<ABC = < ADC
BCA
ASA
Definition
Prove the following
A
B
C
HYP: diagram
Statement
< BAD=<ACD
< ABC = <ABD
ABD˜ CBA
Authorities
HYP
Reflex
AA
AB = BD = AD DEFN
D CB BA CA
AB2 = BC • BD Cross Multipln
CONC: AB2 = BC • BD
Do #5 & 6 
Tips for Success
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Always work on what you know
The more facts you put into a question the
closer you will get to the answer
Extend the lines
Exam Questions & Practice
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We will do more examples on the board
together…
P262, p266, 267, 268, p272, 274
Study Guide
Test 