2.5 Reasoning in Algebra and Geometry

Objective:To connect reasoning in algebra and

geometry

Properties of Equality• Let a, b, and c be any real number.1. Addition If a = b, then a + c = b + c2. Subtraction If a = b, then a – c = b – c3. Multiplication If a = b, then a ∙ c = b ∙ c4. Division If a = b and c ≠ 0, then a/c = b/c

5. Reflexive a = a 6. Symmetric If a = b, then b = a. 7. Transitive If a = b and b = c, then a = c.

8. Substitution If a = b, then b can replace a in any expression

9. Distributive a(b + c) = ab + ac a(b – c) = ab – ac

Justify Steps When Solving

O

M

CA x°(2x + 30)°• What is the value of x?

and Angles that form a linear are supplementary pair are supplementary + = 180 Definition of suppl. Angles (2x + 30) + x = 180 Substitution Property 3x + 30 = 180 Distributive Property

3x = 150 Subtraction Prop. of Eq.X = 50 Division Prop. of Eq.

Try again.

• What is the value of x?• Given: bisects

BR

NA

x° (2x – 75)°

Extra Practice

• What is the value of x?D

F

EC x°(2x – 15)°

Equality and Congruence• Reflexive Property

• Symmetric Property– If , then – If , then

• Transitive Property– If and , then – If and , then – If and , then

Using Equality and Congruence

• What property of equality or congruence is used to justify going from the first statement to the second statement?

A. 2x + 9 = 192x = 10

B. and

C.

Proof

• Proof – convincing argument that uses deductive reasoning; logically shows why a conjecture is true

• Two-column proof – lists each statement on the left and the justification/reason on the right

Here we go…

Given: Prove: 1. What do we know?2. What do we need to do?3. What is our plan?

1. Given

2. Reflexive Prop of =

3. 2 Addition Prop of =

4. Angle Add. Post. 5. Substitution Prop

Again…

Given: Prove: A B C D

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