1. Geometry Proofs
Math 416

2. Time Frame
Definition
Congruent Triangles
Axiom & Proofs
Propositions

3. Definitions 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.

4. Definitions (con’t) 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

5. Gaining Knowledge Proposition Enlightenment Proposition Proposition
Axiom
Definition
Thoerem

6. Definitions
Definition: You define something once you identify its essential characteristics
For example, triangle – a two dimensional polygon with three sides
Must
Not

7. Axiom Axioms: An obvious statement that is acceptable without proof
For example, the shortest distance between two points is a straight line

8. Propositions 1 3 2 Propositions are statements that require proof
Once proven they are called theorems
For example
Proof 1
STATEMENT AUTHORITIES 3
<1 + <3 = 180°
DEFINITION 2
DEFINITION
<2 + < 3 = 180°
<1 = <2 = 180
ALGEBRA

9. Theorums 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

10. 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?

11. Example #2 of a Definition
Stencil #1 
Are Gatus
Which of the following is a Gatu?
Definition: A Gatu is a shape with at least one curved side

12. Axioms 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

13. Axioms
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 =
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 c

14. Postulates ABC YZX By S S S
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 YZX
By S S S B C Y Z

15. Postulates ABC ZXY ° By SAS °
Hypothesis: Given two triangles with two corresponding sides equal and the contained angle equal
Conclusion: The two triangles are congruent X A
ABC ZXY
By SAS Y Z B C

16. Postulates A X ABC ZXY C Y B Z Do #2 
Hypothesis: Given two triangles with two corresponding angles equal and the contained side equal
Conclusion: The two triangles are congruent A X
ABC ZXY O
By ASA O X C Y B X Z
Do #2 

17. Theorems Once again we will not prove But you may be required to
You should be able to

18. Theorems The 90° completion theorem or the complementary angle theorem
HYP: Diagram
CONC < X + <Y = 90° x y
HYP Diagram
CONC <x + <y = 180 x y

19. Vertically Opposite Angle Theorem1 4 3 2
< 1 = < 2
< 3 = <4
Conclusion

20. Triangle Sum Theorem 1 3 2
Conclusion
<1 + <2 + <3 = 180°

21. Isosceles Triangle Theorem
Given an isosceles triangle, the angles opposite the equal sides are equal 1 2
Conclusion
<1 = <2

22. Isosceles Triangle Theorem Converse
Given an isosceles triangle, the sides opposite the equal angles are equal A C B
Conclusion
AB = AC

23. Parallel Line Theorem 1 2 3 4 a b c d <1 < a <2 = <b
Note: The converse is true also to prove // lines
Parallel Line Theorem 1 2
Sometimes called Corresponding angles 3 4 a b c d
<1 < a
<2 = <b
<3 = < c
<4 = <d
Conclusion
<4 = < a
< 3 < b
<3 + <a = 180°
<4 + <b = 180°

24. Parallelogram Theorem and Converse
Conclusion: D A
AD = BC
AB = DC
Opposite Sides
< BAD = <DCB
< ABC = < ADC
Opposite Angles x B C
BX = XD
AX = XC
Diagonals Bisected
In a parallelogram opposite sides are equal, opposite angles are equal and the diagonals bisect each other

25. Triangle Parallel Similarity TheoremC B
Conc
ABC ˜ ADE D E
Do #3 

26. Test Question
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

27. Pythagoras Theorem A b c HYP: Diagram CONC: b2 = a2 + c2 B C a
Given a right angle triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides b c
HYP: Diagram
CONC: b2 = a2 + c2 B C a

28. Pythagoras Examples 202 = x2 + x2 400 = 2x2 200 = x2 14.14 = x2
Solve for x
Solve for x
202 = x2 + x2
400 = 2x2
200 = x2
14.14 = x2
x = 14.14
x2 =
x2 =
x = 10 x 6 20 8 x x

29. The Theorem A
The side opposite the 30° angle is half the hypotenuse.
60° b c
HYP: Diagram
CONC: c = ½ b OR
b = 2c
30° C B

30. The 30-60-90 Theorem ConverseA
If the hypotenuse is twice the length of one of the legs, the angle opposite the leg is 30° 2b b
HYP: Diagram
CONC: <ACB = 30° C B

31. 30-60-90 Examples (2x)2=x2+196 4x2=x2+196 3x2=196 Opposite the 30°
It is half the hypotenuse
x = 12 x 6 14
30° x

32. Exam Question D A Hyp: Diagram Conc: < ABC = < ADC B
Construction AC C

33. Exam Questions Con’t Statement Authorities < DAC = <ACB
Fill in the missing authorities
Statement
Authorities
< DAC = <ACB
< DCA = <BAC
AC = AC
Thus DAC BCA
<ABC = < ADC
// Line Theorum
Reflex
ASA
Definition

34. Prove the following Statement Authorities A B D C HYP: diagram
< BAD=<ACD
HYP
< ABC = <ABD
Reflex
ABD˜ CBA AA
AB = BD = AD
CB BA CA
DEFN B D C
HYP: diagram
CONC: AB2 = BC • BD
AB2 = BC • BD
Cross Multipln
Do #5 & 6 

35. Tips for Success 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

36. Exam Questions & Practice
We will do more examples on the board together…
P262, p266, 267, 268, p272, 274
Study Guide
Test 