Properties from Algebra Section 2-2: Properties from Algebra

a + c = b + d Ex: If x = 12, then x + 2 = 14. Properties of Equality Addition Property If a = b and c = d, then _________________________.   Subtraction Property If a = b and c = d, then __________________________. Multiplication Property If a = b, then _________________________________. Division Property If a = b and c ≠ 0, then ________________________. a + c = b + d Ex: If x = 12, then x + 2 = 14. If x – 3 = 7, then x = 10. a – c = b – d Ex: If x + 2 = 9, then x = 7. ca = cb Ex: If 𝑥 3 = 9, then x = 27. 𝑎 𝑐 = 𝑏 𝑐 Ex: If 4x = 28, then x = 7.

Substitution Property If a = b, then either a or b may be ____________ for the other in any equation (or inequality). substituted Ex: If m∠A = 30° and m∠A = m∠C, then m∠C = 30°. Ex: If 2x + 3 = y and x = 5, then 13 = y. Ex: If x = y and z = y, then x = z.

Reflexive Property a = _____   Symmetric Property If a = b, then ____________________. Transitive Property If a = b and b = c, then _____________. a Ex: If m∠A, then m∠A. b = a Ex: If AB = CD, then CD = AB. a = c Ex: If m∠A = m∠B and m∠B = m∠C, then m∠A = m∠C.

𝐷𝐸 ∠D 𝐹𝐺 ≅ 𝐷𝐸 ∠E ≅ ∠D 𝐷𝐸 ≅ 𝐽𝐾 ∠D ≅ ∠F Properties of Congruence Reflexive Property 𝐷𝐸 ≅ ________ ∠D ≅ _________ Symmetric Property If 𝐷𝐸 ≅ 𝐹𝐺 , then _____________________________. If ∠D ≅ ∠E, then ______________________________. Transitive Property If 𝐷𝐸 ≅ 𝐹𝐺 and 𝐹𝐺 ≅ 𝐽𝐾 , then ­­­_______________. If ∠D ≅∠E and ∠E ≅∠F, then _________________. 𝐷𝐸 ∠D 𝐹𝐺 ≅ 𝐷𝐸 ∠E ≅ ∠D 𝐷𝐸 ≅ 𝐽𝐾 ∠D ≅ ∠F

Properties of Real Numbers Commutative Property a + b = __________, ab = _______   Associative Property a + (b + c) = _________, a(bc) = ________ Distributive Property a(b + c) = __________ b + a ba (a + b) + c (ab)c ab + ac

Examples: Justify each step with a “Property from Algebra.” Follow the example below:   Given: 4x – 5 = –2 Prove: x = 3 4 Statements Reasons 1. 4x – 5 = –2 1. Given 2. 4x = 3 2. Addition Property of Equality 3. x = 3 4 3. Division Property of Equality

Multiplication Property of Equality 1. Given: 3𝑎 2 = 6 5 Prove: a = 4 5   Statements Reasons 1. 3𝑎 2 = 6 5 1. Given 2. 3a = 12 5 2. 3. a = 4 5 3. Multiplication Property of Equality Division Property of Equality

2. Given: 𝑧+7 3 = –11 Prove: z = –40 Statements Reasons 1 2. Given: 𝑧+7 3 = –11 Prove: z = –40 Statements Reasons 1. 𝑧+7 3 = –11 1. Given 2. z + 7 = –33 2. 3. z = –40 3. Multiplication Property of Equality Subtraction Property of Equality

Addition Property of Equality 3. Given: 15y + 7 = 12 – 20y Prove: y = 1 7 Statements Reasons 1. 15y + 7 = 12 – 20y 1. Given 2. 35y + 7 = 12 2. 3. 35y = 5 3. 4. y = 1 7 4. Addition Property of Equality Subtraction Property of Equality Division Property of Equality

Multiplication Property of Equality 4. Given: x – 2 = 2𝑥+8 5 Prove: x = 6 Statements Reasons 1. x – 2 = 2𝑥+8 5 1. Given 2. 5(x – 2) = 2x + 8 2. 3. 5x – 10 = 2x + 8 3. 4. 3x – 10 = 8 4. 5. 3x = 18 5. 6. x = 6 6. Multiplication Property of Equality Distributive Property Subtraction Property of Equality Addition Property of Equality Division Property of Equality

Substitution Property Practice   1. a = b + c 1. Given d = e + f 2. a = d 2. Given 3. ____________ 3. Substitution b + c = e + f

II. 1. a = b + c 1. Given d = e + f 2. b + c = e + f 2. Given 3. _____________ 3. Substitution a = d

(Diagram is for III. And IV.) III. 1. DF = AC 1. Given       1. DF = AC 1. Given 2. DE + EF = ____ 2. ____________ AB + BC = ____ ____________ 3. ____________ 3. Substitution • • • D E F • • • A B C DF Segment AC Addition Post. DE + EF = AB + BC

IV. 1. DE + EF = AB + BC 1. Given 2. DE + EF = _____ 2 IV. 1. DE + EF = AB + BC 1. Given 2. DE + EF = _____ 2. ______________ AB + BC = _____ ______________ 3. _____________ 3. Substitution DF Segment AC Addition Post. DF = AC

m∠WOY = m∠1 + m∠2 2. _____________ m∠XOZ = m∠3 + m∠2 _____________ V. 1. ∠WOY  ∠XOZ 1. Given m∠WOY = m∠1 + m∠2 2. _____________ m∠XOZ = m∠3 + m∠2 _____________ 3. ________________ 3. Substitution ________________ Angle Addition Post. m∠1 + m∠2 = m∠3 + m∠2 W • X • Y • 1 2 3 • • O Z

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