1. Algebraic and Geometric Proofs
Chapter 2
Algebraic and Geometric Proofs

2. Section 2.1 – Counterexamples and Algebraic Proofs
Learning Targets:
I can make an educated guess based on inductive reasoning.
I can find counterexamples.
I can use algebra to write two-column proofs.

3. An educated guess

4. An example that shows that your conjecture is not true
Sample: 2 * 3 = 6… Not odd.
TRUE 

5. logical argument
supported
true

6. 20 = 20 a = a If x = 20, then 20 = x. If a = b, then b = a.
If x = y and y = 3, then x = 3.
If a = b and b = c, then a = c.
If x = 5, then x + 2 = 5 + 2
If a = b, then a + c = b + c
If x = 5, then x – 2 = 5 – 2
If a = b, then a – c = b – c

7. If a = b, then ac = bc
If x = 5, then 2x = 2(5)
If a = b, then a/c = b/c (c cannot equal 0!!!)
If x = 5, then x/2 = 5/2
If a = b, then a can be replaced by b in ANY equation!
If = 0, then 3x – = 3x + 0
VERY IMPORTANT!!!
a(b + c) = ab + ac
4(x + 3) = 4x + 12

8. statements
reasons
Each step of the proof (conjectures)
Properties that justify each step (definitions, theorems, postulates)

9. 3x + 5 = 17
Given
3x + 5 – 5 = 17 – 5
Subtraction Property
3x = 12
Substitution Property
3x/3 = 12/3
Division Property
x = 4
Substitution Property

10. 6x – 3 = 4x + 1
Given
6x – = 4x
Addition Property
6x = 4x + 4
Substitution Property
6x – 4x = 4x – 4x + 4
Subtraction Property
2x = 4
Substitution Property
2x/2 = 4/2
Division Property
x = 2
Substitution Property

11. Homework Assignment:
2.1 Worksheet  Start the chapter on a good note! What is your goal for this chapter?

12. Warm Up: Conjectures and Counterexamples
Determine whether each conjecture is true or false. Give a counterexample for any false conjecture.
Conjecture
True or False
(circle one)
Counterexample if false 1.
Teenagers are not good drivers.
T F 2.
Teachers never show movies. 3.
The sum of two odd integers (Like 3 + 5) is even. 4.
The product of an odd integer and an even integer (Like 5x2) is even. 5.
The opposite of an integer is a negative integer.
Quiz question

13. 2.2 – Geometric Proof with Congruence
Learning Targets:
I can write proofs involving segment congruence.
I can write proofs involving angle congruence.

14. AB = AB
If AB = CD, then CD = AB
If AB = CD and
CD = EF, then
AB = EF

16. Given
Given
Transitive Property

17. halfway

19. B is the midpoint of AC
Given
Given
C is the midpoint of BD
Midpoint Theorem
Midpoint Theorem
Transitive Property

20. cuts
2 equal pieces
congruent

21. Warm Up
Please complete your ACT questions in Wednesday’s section of your warm up sheet!

24. Warm Up

26. 2.3 Geometric Proofs with Addition
Learning Targets
I can write proofs involving segment addition.
I can write proofs involving angle addition.

27. Given
Given
Segment Addition Postulate (SAP)
Substitution Property

28. straight line
180
right 90

29. Given
Angle Addition Post. (AAP)
Angle Addition Post. (AAP)
Substitution
Def. of congruent angles

30. Warm Up

32. Proof review Work in stations Use flip books only, no notes
Write “reasons” only on your answer sheet
If you have extra time at a station, start working on the problems on the back of your answer sheet

33. Given
Given
Def of Angle Bisector
Def of Angle Bisector
Transitive

34. Given
Addition Property
Substitution
Division Property
Substitution

35. Given
Given
Segment addition
Segment addition
Substitution
Subtraction
Substitution

36. Given
Angle addition
Angle addition
Substitution
Subtraction
Substitution