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

USING PROPERTIES OF EQUALITY AND CONGRUENCE • Subtraction Property of Equality Property Transitive of Congruence Symmetric Property of Equality

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

ALGEBRA PROOF G IVEN: 20 = 3 X + 5 PR OVE: X = 5 Statements Reasons 1. 20 = 3 x + 5 1. Given 2. 15 = 3 x 3. 5 = x 4. x = 5 2. Subtraction Prop. of = 3. Division Property = 4. Symmetric Property =

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 Reasons 1. Given 2. Addition Postulate 3. Substitution Prop. = 4. Simplify 5. Subtraction Prop. = 6. Division 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 C 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 Property

EX 4: 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 =

JUSTIFYING STEPS WHEN SOLVING AN EQUATION • Reasons 1. Given 2. Definition of Angle Bisector 3. Substitution Property of = 4. Subtraction Property of = 5. Division Property of =

JUSTIFYING STEPS WHEN SOLVING AN EQUATION • What is the value of x? Justify each step. Reasons 1. Definition of Supplementary Angles 2. Substitution Property of = 3. Simplify/Combine Like Terms 4. Subtraction Property of = 5. Division Property of =

WRITING A TWO-COLUMN PROOF • Write a two-column proof. Given: Prove:

WRITING A TWO-COLUMN PROOF