Before: What is the converse of the following true conditional? If the converse is also true, combine the statements as a biconditional. If a number is even, then it is divisible by 2. What are the two conditional statements that form this biconditional? A quadrilateral is a trapezoid if and only if it has exactly one pair of parallel sides. Is this definition of a rectangle reversible? If yes, write it as a true biconditional. A rectangle is a parallelogram with four right angles.

Before: A detective uses deductive reasoning to solve a case by gathering, combining, and analyzing clues. How might you use deductive reasoning in geometry?

2.5 - During: Reasoning in Algebra and Geometry I can connect reasoning in algebra and geometry.

Algebraic properties of equality are used in geometry Algebraic properties of equality are used in geometry. They will help you solve problems and justify each step you take. In geometry you accept postulates and properties as true. Some of the properties that you accept as true are the properties of equality from algebra.

Key Concept: Property of Equality 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 a ∙ c = b ∙ c Division Property If a = b and c ≠ 0, then a/c = b/c Reflexive Property a = a Symmetric Property If a = b, then b = a Transitive Property If a = b and b = c, then a = c Substitution Property If a = b, then b can replace a in any expression.

Key Concept: The Distributive Property Use multiplication to distribute “a” to each term of the sum or difference within the parentheses. Sum Difference a(b + c) = ab + bc a(b – c) = ab - bc You use deductive reasoning when you solve an equation. You can justify each step with a postulate, a property, or a definition. For example, you can use the Distributive Property to justify combining like terms.

Problem: Justifying Steps When solving an Equation Find x. <AOM and <MOC are supplementary. Given m<AOM + m<MOC = 180 (2x + 30) + x = 180 3x + 30 = 180 3x = 150 X = 50

Problem: Justifying Steps When solving an Equation Find x. AB bisects <RAN Given <RAB = <NAB x = 2x – 75 -x = -75 x = 75

Problem: Justifying Steps When solving an Equation Find x. m<CDF + m<EDF = 180 x + 2x – 15 = 180 3x – 15 = 180 3x = 195 x = 65

Key Concept: Properties of Congruence Some properties of equality have corresponding properties of congruence. Key Concept: Properties of Congruence Reflexive Property Symmetric Property Transitive Property

Problem: Using Properties of Equality and Congruence What is the name of the property of equality or congruence that justifies going from the first statement to the second statement? 2x + 9 = 19 2x = 10 <O <W and <W <L <O <L m<E = m<T m<T = m<E

Problem: Using Properties of Equality and Congruence What is the name of the property of equality or congruence that justifies going from the first statement to the second statement? 7x + 3 = 24 7x = 21 RS = ST and ST = TU RS = TU 5x = 10 10 = 5x

A proof is a convincing argument that uses deductive reasoning A proof is a convincing argument that uses deductive reasoning. A proof logically shows why a conjecture is true. A two-column proof lists each statement on the left. The justification, or reason for each statement, is on the right. Each statement must follow logically from the steps before it. The diagram below shows the setup for a two-column proof.

Problem: writing a two-Column Proof Write a two-column proof. Given: m<1 = m<3 Prove: m<AEC = m<DEB Statements Reasons m<1 = m<3 m<2 = m<2 m<1 + m<2 = m<3 + m<2 m<1 + m<2 = m<AEC m<3 + m<2 = m<DEB m<AEC = m<DEB

Problem: writing a two-Column Proof Write a two-column proof. Given: AC = BD Prove: AB = CD Statements Reasons AC = BD BC = BC AC = AB + BC BD = BC + CD AB + BC = BC + CD AB = CD

After: Lesson Check Name the property of equality or congruence that justifies going from the first statement to the second statement. m<A = m<S and m<S = m<K m<A = m<K 3x + x + 7 = 23 4x + 7 = 23 4x + 5 = 17 4x = 12

After: Lesson Check (continued) Fill in the reasons for this algebraic proof. Given: 5x + 1 = 21 Prove: x = 4 Statements Reasons 5x + 1 = 21 5x = 20 x = 4

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