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Connecting Algebraic Reasoning to Geometric Properties

In this lesson, we aim to connect algebraic reasoning with geometric principles, focusing on the Properties of Equality. The key properties, including Reflexive, Symmetric, Transitive, Addition, Subtraction, Multiplication, and Division, are explored in detail. We demonstrate how these properties apply to angle and segment relationships through various examples and an exercise on solving for unknowns. Engaging in this integration of algebra and geometry helps solidify understanding and facilitates problem-solving skills in mathematical contexts.

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Connecting Algebraic Reasoning to Geometric Properties

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  1. Chapter 2 Lesson 4 Objective: To connect reasoning in algebra to geometry.

  2. Properties 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

  3. If point B is in the interior of AOC, then m AOB + m BOC = m AOC. • B • O The Distributive Property a(b+c) = ab + ac Angle Addition Postulate • A C

  4. • B • Given: m AOC = 139 A • O C m AOB + m BOC = m AOC Example 1: Solve for x and justify each step. x° (2x + 10)° Angle Addition Postulate x + 2x + 10 = 139 Substitution Property 3x + 10 = 139 Simplify 3x = 129 Subtraction Property of = x = 43 Division Property of =

  5. Example 2: Justify each step used to solve 5x – 12 = 32 + x for x. 5x = 44 + x 4x = 44 X = 11 Addition Property of Equality Subtraction Property of Equality Division Property of Equality

  6. • M • K (2x + 40)° • 4x° LM bisects KLN Given m MLN = m KLM Definition of angle bisector 4x = 2x + 40 _____________________ 2x = 40 _____________________ x = 20 _____________________ L N Example 3: Fill in each missing reason. Substitution Prop. Subtraction Prop. of Equality Division Prop. Of Equality

  7. 2y 3y-9 A B C Example 4: Solve for y and justify each step. Given: AC = 21 AB + BC = AC 2y + (3y – 9) = 21 5y – 9 = 21 5y = 30 Y = 6 Segment Addition Postulate Substitution Property Simplify Addition Property of Equality Division Property of Equality

  8. Properties of Congruence Reflexive Property AB AB A A Symmetric Property If AB CD, then CD AB If A B, then B A Transitive Property If AB CD and CD EF, then AB EF If A B and B C, then A C.

  9. a. K K d. If RS TW and TW PQ, then RS PQ. Example 5: Name the property of equality or congruence that justifies each statement. Reflexive Property of Congruence b. If 2x – 8 = 10, then 2x = 18 Addition Property of Equality c. If x = y and y + 4 = 3x, then x + 4 = 3x. Substitution Property of Equality Transitive Property of Congruence

  10. Homework Page 91-93 #1-30

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