1. Proofs
A proof is a written account of the complete thought process used to reach a conclusion.
Day 1

2. Algebraic Proofs Algebraic Proofs use these Properties: Addition
Subtraction
Multiplication
Division
Combining Like Terms
Distributive
Add to both sides of =
Subtract from both sides of =
Multiply to both sides of =
Divide from both sides of =
Combine on one side of =
3(x + 4)  3x + 12

3. Multiplication Substitution Addition Distributive Combine Like Terms
State the property that justifies each statement.
If 𝑥 2 =7, then x = ____________________________
If x = 5 and b = 5, then x = b. ________________________
If XY – AB = WZ – AB, then XY = WZ. __________________
4. If 2(x + 3) = 12, then 2x + 6 = 12. ____________________
If 3x + 4x = 7, then 7x = 7. __________________________
If 4x = 16, then x = 4. ______________________________
If 3x + 6 = 12, then 3x = 6 _______________________________
Multiplication
Substitution
Addition
Distributive
Combine Like Terms
Division
Subtraction

4. Two-Column Proof
Contains statements and reasons organized into two columns
Given:
Prove:
Is the first step in the proof
Is the last step in the proof

5. 8. Algebraic Proof Given: 3(5x + 3) + 2 = -19 Prove: x = -2
Statements Reasons
1. 3(5x + 3) + 2 = Given
x = -19
2. Distributive
x + 11 = -19
3. Combine Like Terms
x = -30
4. Subtraction
x = -2
5. Division

6. Reflexive, Symmetric, and Transitive
Three properties used in Proofs
Reflexive
Symmetric
Transitive
If a, then a = a
If a = b, then b = a
If a = b, b = c, then a = c

7. 9. Reflexive Property Example
Reflexive : Equal to Itself
Statements Reasons T
1. T Given
2. T  T
2. Reflexive Property
Statements Reasons A B
1. 𝐴𝐵 Given
2. 𝐴𝐵  𝐴𝐵
2. Reflexive Property

8. 10. Symmetric Property Symmetric : Switch Sides Statements Reasons
1. M  Y Given
2. Y  M
2. Symmetric Property
Statements Reasons
1. 𝐴𝐵  𝐶𝐷 Given
2. 𝐶𝐷  𝐴𝐵
2. Symmetric Property

9. 11. Transitive Property Example
Transitive : If a = b and b = c, then a = c
Statements Reasons
1. A  B; B  C Given
2. A  C
2. Transitive Property A B C

10. 12. Substitution Property
Substitution: If a = b, then a can be replaced by b
in any equation or expression
Statements Reasons C D
1. AB = CD; Given
2. AB + BC = AC
2. Segment Addition A B C
3. CD + BC = AC
3. Substitution
4. AB + BC = CD + BC
4. Substitution

11. 13 S N M R P Q
Given: 𝑀𝑁  𝑃𝑄
𝑃𝑄  𝑅𝑆
Prove: 𝑅𝑆  𝑀𝑁
Statements
Reasons
1. 𝑀𝑁  𝑃𝑄 ; 𝑃𝑄  𝑅𝑆
1. ____________
Given
MN = PQ; PQ = RS
2. __________________
2. Def. of Congruent Segments
3. __________________
MN = RS
3. Transitive
4. RS = MN
4. ________________________
Symmetric
5. 𝑅𝑆  𝑀𝑁
Def. of Congruent Segments
5. ________________________

12. 14
Given: L is the midpoint of 𝐾𝑀
𝐿𝑀  𝐴𝐵
Prove: 𝐾𝐿  𝐴𝐵 B L K A M
Statements
Reasons
1. L is the midpoint of 𝐾𝑀
1. ____________
Given
2. __________________
𝐾𝐿  𝐿𝑀
2. _________________
Def. of Midpoint
3. 𝐿𝑀  𝐴𝐵
3.__________________
Given
4. _____________
𝐾𝐿  𝐴𝐵
4. __________________
Transitive

13. 3
Given: 𝐴𝐶  𝐵𝐷
Prove: 𝐴𝐵  𝐶𝐷 D B A C
Statements
Reasons
1. 𝐴𝐶  𝐵𝐷 ;
1. ____________
Given
2. __________________
AC = BD
2. Def. of Congruent Segments
3. __________________
__________________
AC = AB + BC
3. Segment Addition
BD = BC + CD
4. AB + BC = BC + CD
4. ________________________
Substitution
5. AB = CD
Subtraction
5. ________________________
6. _________________
𝐴𝐵  𝐶𝐷
6. ________________________
Def. of Congruent Segments

14. SEGMENT PROOF ACTIVITY
Teams of Two
Fill in the missing statements and reasons
One Person does one proof and then you switch
Help each other