1. Starter(s):
Find one counterexample to show that each conjecture is false.
All vehicles on the highway have exactly four wheels.
2. All states in the United States share a border with another state.
3. All plurals end with the letter s.
4. The difference between two integers is always positive.
5. All pentagons have exactly five congruent sides.
6. All numbers that are divisible by 3 are also divisible by 6.
7. All whole numbers are greater than their opposites.
8. All prime numbers are odd integers.

2. Topic: Algebraic Proofs Section: 2-5
Then/Now

4. Concept

5. ? ? ? ?

8. 2(5 – 3a) – 4(a + 7) = 92 Original equation
Justify Each Step When Solving an Equation
Solve 2(5 – 3a) – 4(a + 7) = 92.
2(5 – 3a) – 4(a + 7) = 92 Original equation
10 – 6a – 4a – 28 = 92 Distributive Property
–18 – 10a = 92 Substitution Property
–18 – 10a = Addition Property
Example 1

9. –10a = 110 Substitution Property
Justify Each Step When Solving an Equation
–10a = 110 Substitution Property
Division Property
a = –11 Substitution Property
Answer: a = –11
Example 1

10. ? Solve –3(a + 3) + 5(3 – a) = –50. A. a = 12 B. a = –37 C. a = –7
D. a = 7
Example 1

11. Begin by stating what is given and what you are to prove.
Write an Algebraic Proof
Begin by stating what is given and what you are to prove.
Example 2

12. 2. Subtraction Property of Equality
Write an Algebraic Proof
Statements Reasons
Proof:
1. Given 1.
d = 20t + 5
2. d – 5 = 20t
2. Subtraction Property of Equality 3.
3. Division Property of Equality
= t 4.
4. Symmetric Property of Equality
Example 2

13. Which of the following statements would complete the proof of this conjecture?
If the formula for the area of a trapezoid is , then the height h of the trapezoid is given by
Example 2

14. 3. Division Property of Equality
Statements Reasons
Proof: 3.
3. Division Property of Equality 4.
4. Symmetric Property of Equality
1. Given 1.
2. _____________
2. Multiplication Property of Equality ?
Example 2

15. A. 2A = (b1 + b2)h B. C. D.
Example 2

16. Write a Geometric Proof
If A B, mB = 2mC, and mC = 45, then mA = 90. Write a two-column proof to verify this conjecture.
Example 3

17. 3. Transitive Property of Equality 3. mA = 2mC
Write a Geometric Proof
Statements Reasons
Proof:
1. Given 1.
A B; mB = 2mC; mC = 45
2. mA = mB
2. Definition of angles
3. Transitive Property of Equality
3. mA = 2mC
4. Substitution 4.
mA = 2(45)
5. mA = 90
5. Substitution
Example 3

18. Example 3

19. 3. Definition of congruent segments
Statements Reasons
Proof:
1. Given 1. 2.
2. _______________ ?
3. AB = RS
3. Definition of congruent segments
4. AB = 12
4. Given
5. RS = 12
5. Substitution
Example 3

20. A. Reflexive Property of Equality B. Symmetric Property of Equality
C. Transitive Property of Equality
D. Substitution Property of Equality
Example 3