2.5 and 2.6 Properties of Equality and Congruence

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Presentation transcript:

2.5 and 2.6 Properties of Equality and Congruence

Objective: You will use deductive reasoning to: Write proofs using geometric theorem To use algebraic properties in logical arguments.

Algebraic Properties Substitution Property: If a = b, then a can be substituted for b in an equation or expression. Distributive Property: a(b + c) = ab + ac, where a, b, and c are real numbers.

Algebraic Properties Addition Property: If a = b, then a + c = b + c. Subtraction Property: If a = b, then a – c = b – c. Multiplication Property: If a = b, then ac = bc. Division Property: If a = b and c = 0, then a/c = b/c.

Example 1: Write a two-column proof to solve the equation. Statements Reasons 3x + 2 = 8 3x + 2 – 2 = 8 – 2 3x = 6 3x ÷ 3 = 6 ÷ 3 x = 2 1. Given 2. Subtraction Prop 3. Simplify 4. Division Prop 5. Simplify

Example 2: Write a two-column proof to solve the equation. Statements Reasons 4x + 9 = 16 – 3x 4x + 9 + 3x = 16 – 3x + 3x 7x + 9 = 16 7x + 9 – 9 = 16 – 9 7x = 7 7x ÷ 7 = 7 ÷ 7 x = 1 Given Addition Prop Simplify Subtraction Prop Division Prop

Example 3: Write a two-column proof to solve the equation. Statements Reasons 2(-x – 5) = 12 Given -2x – 10 = 12 Distributive Prop -2x – 10 + 10 = 12 + 10 Addition Prop -2x = 22 Simplify -2x ÷ -2 = 22 ÷ -2 Division Prop x = -11 Simplify

Reflexive Property Equality Congruence

Symmetric Property Equality: Congruence:

Transitive Property Equality: Congruence:

Properties of Equality Addition Property: adding a number to each side of an equation Subtraction Property: subtracting a number from each side of an equation

Properties of Equality Multiplication Property: multiplying by a number on each side of an equation Division Property: dividing by a number on each side of an equation Substitution Property: substituting a number for a variable in an equation to produce an equivalent equation

Definitions Theorem: A true statement that follows as a result of other true statements. Two-column proof: Most commonly used. Has numbered statements and reasons that show the logical order of an argument.

Use the diagram and the given information to complete the missing steps and reasons in the proof. GIVEN: LK = 5, JK = 5, JK ≅ JL PROVE: LK ≅ JL Statements: Reasons: _______________ LK = JK LK ≅ JK JK ≅ JL ________________ Given Transitive Property _______________