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Understanding Algebraic Properties of Equality in Geometry

This chapter explores essential algebraic properties of equality and their application in geometry, particularly in proving statements about segments and angles. Key concepts include the Addition, Subtraction, Multiplication, and Division properties, along with the Distributive property and the Substitution property. The chapter also covers logical arguments in proofs, including two-column proofs that structure statements and their reasons systematically. Examples demonstrate the practical use of these properties in proving congruence and relationships between segments and angles.

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Understanding Algebraic Properties of Equality in Geometry

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  1. GEOMETRY: CHAPTER 2 Ch. 2.4 Reason Using Properties from Algebra Ch. 2.5: Prove Statements about Segments and Angles

  2. 2.4: KEY CONCEPT: Algebraic Properties of Equality Let a, b, and c be real numbers. 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.

  3. Substitution Property: If a=b, then a can be substituted for b in any equation or expression. Distributive Property: a (b + c ) = ab + ac.

  4. Ex. 1 Solve: Write reasons for each step

  5. Ex. 2: Use the Distributive Property and Write reasons for each step. Solution: Equation Explanation Reason -2 (3x + 1)=40 Write original equation Given -6x -2 = 40 Multiply Distribution Property -6x = 42 Add 2 to each side Addition Property of Equality x= -7 Divide each side by -6 Division Property of Equality

  6. Key Concepts: Reflexive Property of Equality: Real Numbers: for any real number a, a=a. Segment Length: for any segment AB, AB=AB. Angle Measure: for any angle A,

  7. Symmetric Property of Equality

  8. Transitive Property of Equality

  9. Ex. 3: The city is planning to add two stations between the beginning and end of a commuter train line. Use the information given. Determine whether RS=TU. Image taken from: Geometry. McDougal Littell: Boston, 2007. P. 107.

  10. Ex. 3 (cont.) Given: RT=SU Prove: RS=TU EQUATION REASON RT=SU Given ST=ST Reflexive Property RT-ST=SU-ST Subtraction Property of Equality RT-ST=RS Segment Addition Postulate SU-ST=TU Segment Addition Postulate RS=TU Substitution Property of Equality

  11. For more examples, go to CH. 2, Lesson 5, Example 4 http://www.classzone.com/cz/books/geometry_2007_na/get_chapter_group.htm?cin=2&rg=help_with_the_math&at=powerpoint_presentations&var=powerpoint_presentations

  12. 2.5 Prove Statements about Segments and Angles A proof is a logical argument that shows a statement is true. A two-column proof has numbered statements and corresponding reasons that show an argument in a logical order.

  13. In a two-column proof, each statement in the left-hand column is either given information or the result of applying a known property or fact to statements already made. The explanation for the corresponding statement is in the right-hand column.

  14. Image taken from: Geometry. McDougal Littell: Boston, 2007. P. 112. Ex. 5 You are designing a logo to sell daffodils. Use the information given. Determine whether the measure of angle EBA is equal to the measure of angle DBC.

  15. Image taken from: Geometry. McDougal Littell: Boston, 2007. P. 112. Ex. 5 (cont.)

  16. Theorems—The reasons used in a proof can include definitions, properties, postulates, and theorems. A theorem is a statement that can be proven. Once you have proven a theorem, you can use the theorem as a reason in other proofs.

  17. Theorem 2.1 Congruence of Segments. Segment congruence is reflexive, symmetric, and transitive.

  18. Theorem 2.2 Congruence of Angles Angle congruence is reflexive, symmetric, and transitive.

  19. Ex. 6: Name the property illustrated by the statement. • Transitive Property of Angle Congruence. • Symmetric Property of Segment Congruence.

  20. Ex. 7 Use properties of equality Prove this property of midpoints: If you know that M is the midpoint of segment AB, prove that AB is two times AM and AM is one half of AB. GIVEN: M is the midpoint of segment AB. Prove: a. AB=2 (AM) b. AM= ½ AB

  21. Ex. 7 (cont.) STATEMENTS REASONS

  22. Turn to page 103 in your book and look at the bottom right example. Refer back to this page for writing proofs!

  23. Ex. 8 Go To Chapter 6, Lesson 2, Example 4: http://www.classzone.com/cz/books/geometry_2007_na/get_chapter_group.htm?cin=2&rg=help_with_the_math&at=powerpoint_presentations&var=powerpoint_presentations Write this example in your notes as Ex. 8.

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