UNIT 2 REASONING IN ALGEBRA AND GEOMETRY REASONING
UNIT 2 REASONING IN ALGEBRA AND GEOMETRY
REASONING IN ALGEBRA AND GEOMETRY • Algebraic properties of equality are used in Geometry. • Will help you solve problems and justify each step. • In Geometry, you accept postulates and properties as true. • Some of the properties you accept as true are the properties of equality from Algebra.
PROPERTIES OF EQUALITY •
DISTRIBUTIVE PROPERTY • Use multiplication to distribute a to each term of the sum or difference within the parentheses. Sum: a (b + c) = ab + ac Difference: a (b – c) = ab – ac
PROPERTIES OF CONGRUENCE
PROOF • A proof – a convincing argument that uses deductive reasoning. • Logically shows why a conjecture is true. • A two-column proof lists each statement on the left and the reasons on the right. • Each statement must follow logically from the steps before it.
PROOF
Let’s Watch A Short Video! What is a Proof?
JUSTIFYING STEPS WHEN SOLVING AN EQUATION • What is the value of x? Justify each step.
JUSTIFYING STEPS WHEN SOLVING AN EQUATION Angles that form a linear pair are suppl. Definition of supplementary angles Substitution Property Simplify Subtraction Property of Equality Division Property of Equality
USING PROPERTIES OF EQUALITY AND CONGRUENCE • Subtraction Property of Equality Property Transitive of Congruence Symmetric Property of Equality
WRITING A TWO-COLUMN PROOF • Write a two-column proof. Given: Prove:
WRITING A TWO-COLUMN PROOF
EXAMPLE #2 GIVEN: M AOC = 139 PROVE: X = 43 A B x (2 x +10) C O Statements 1. m AOC = 139, m AOB = x, m BOC = 2 x + 10 2. m AOC = m AOB + m BOC 3. 139 = x + 2 x + 10 4. 139 = 3 x + 10 5. 129 = 3 x 6. 43 = x 7. x = 43 1. 2. 3. 4. 5. 6. 7. Reasons Given Addition Prop. Substitution Prop. Simplify Subtraction Prop. Division Prop. Symmetric Prop.
EXAMPLE 3: SEGMENT ADDITION PROOF GIVEN: AB = 4 + 2 X A 15 – x BC = 15 – X 4 + 2 x B AC = 21 PROVE: X = 2 Statements 1. AB=4+2 x, BC=15 – x, AC=21 Reasons 1. Given 2. AC = AB + BC 2. Segment Addition Postulate 3. 21 = 4 + 2 x + 15 – x 3. Substitution Property 4. 21 = 19 + x 4. Simplify/Combined Like Terms 5. 2 = x 5. Subtraction C
EX: GIVEN: PQ=2 X+5 QR=6 X-15 PR=46 P PROVE: X=7 Statements 1. PQ=2 x+5, QR=6 x-15, PR=46. 2. PQ+QR=PR 3. 2 x+5+6 x-15=46 4. 8 x-10=46 5. 8 x=56 6. x=7 R Q Reasons 1. Given 2. Segment Add. Postulate 3. Substitution Prop. of = 4. Simplify 5. Addition Prop. of = 6. Division Prop. of =
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