section 2-4: reasoning in algebra tpi 32a: apply reflective, transitive, or symmetric prooperties of...
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Section 2-4: Reasoning in AlgebraTPI 32A: apply reflective, transitive, or symmetric prooperties of equality or congruence
Objectives:• Connect reasoning in algebra and geometry• Justify steps in deductive reasoning
In geometry • postulates, definitions, & properties are accepted as true• you use deductive reasoning to prove other statements
We will look at some basic properties used to justify statements…..
….. which leads to writing proofs.
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Addition Property of Equality
If a = b, then a + c = b + c Add same amount to both sides of an equation.
Subtraction Property of Equality
If a = b, then a - c = b - c Subtract same amount to both sides of an equation.
Multiplication Property of Equality
If a = b, then a ∙ c = b ∙ c Multiply both sides of an equation by the same amount.
Division Property of Equality
If a = b and c 0, then
Divide both sides of an equation by the same amount.
a b
c c
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Reflective Property of Equality
a = a Ex: 5 = 5
Symmetric Property of Equality
If a = b, then b = a Ex: 3 = 2 + 1 and 2 + 1 = 3 are the same.
Transitive Property of Equality
If a = b and b = c, then a = c.EX: If 3 + 4 = 7 and 5 + 2 = 7, then 3 + 4 = 5 + 2.
Substitution Property of Equality
If a = b , then b can replace a in any expression. Ex: a = 3; If a = b, then 3 = 3.
Distributive Property
a(b + c) = ab + ac Ex: 3(x + 3) = 3x + 9
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Using Properties to Justify Steps in Solving Equations
Algebra Solve for x and justify each step.
Given: m AOC = 139
m AOC = 139 Given
m AOB + m BOC = m AOC Angle Addition Postulate
x + 2x + 10 = 139 Substitution Property
3x + 10 = 139 Simplify
3x = 129 Subtraction Property of Equality
x = 43 Division Property of Equality
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Using Properties to Justify Steps in Solving Equations
Solve for x and justify each step.
Given: LM bisects KLN
LM bisects KLN Given
MLN = KLM
4x = 2x + 40
2x = 40
x = 20
Def of Angle Bisector
Substitution Property
Subtraction Property of Equality
Division Property of Equality
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Using Properties to Justify Steps in Solving Equations
Solve for y and justify each step
Given: AC = 21
AC = 21 Given
AB + BC = AC Segment Addition Postulate
2y + 3y - 9 = 21 Substitution Property
5y – 9 = 21 Simplify
5y = 30 Addition Property of Equality
y = 6 Division Property of Equality
Find AB and BC by substituting y = 6 into the expressions.
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Reflective Property of Congruence
AB ABA A
Symmetric Property of Congruence
Transitive Property of Congruence
If AB CD and AB EF, then CD EF.If A B and B C, then A C.
The Reflective, Symmetric, and Transitive Properties of Equality have corresponding properties of congruence that can be used to justify statements.
If AB CD, then CD AB.If A B, then B A
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Using Properties of Equality and Congruence
Name the property of congruence or equality the justifies each statement.
a. K K Reflective Property of
Symmetric Property of Equality
b. If 2x – 8 = 10, then 2x = 18 Addition Property of Equality
c. If RS TW and TW PQ,
then RS PQ.
Transitive Property of
d. If m A = mB, then
m B = mA
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Use what you know about transitive properties to complete the following:
The Transitive Property of Falling Dominoes:
If domino A causes domino B to fall, and domino B causes domino C to fall, then domino A causes domino _______ to fall.C