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GEOMETRY: CHAPTER 2 Ch. 2.4 Reason Using Properties from Algebra Ch. 2.5: Prove Statements about Segments and Angles 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. Substitution Property: If a=b, then a can be substituted for b in any equation or expression. Distributive Property: a (b + c ) = ab + ac. Ex. 1 Solve: Write reasons for each step 2x + 4 = 10 – 4x Write original equation. Given 2x +4 +4x = 10 - 4x+4x Add 4x to each Addition Property side. of Equality 6x + 4 = 10 Combine Like terms. Simplify. 6x = 6 Subtract four Subtraction from each side. Property of Equality x=1 Divide each side by 6. Division Property of Equality 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 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, mA mA. Symmetric Property of Equality Real Numbers: For any real numbers a and b, if a b, then b a. Segment Length: For any segments AB and CD, if AB CD, then CD AB. Angle Measure: For any angles A and B, if mA mB, then mB mA. Transitive Property of Equality Real Numbers: For any real numbers a, b and c, if a b and b c, then a c. Segment Length: For any segments AB, CD and EF , if AB CD and CD EF , then AB EF . Angle Measure: For any angles A, B and C , if mA mB and mB mC , then mA mC. 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. Given: RT=SU Prove: RS=TU EQUATION RT=SU ST=ST RT-ST=SU-ST RT-ST=RS SU-ST=TU RS=TU Ex. 3 (cont.) REASON Given Reflexive Property Subtraction Property of Equality Segment Addition Postulate Segment Addition Postulate Substitution Property of Equality 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_presentation s 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. 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. Ex. 5 Image taken from: Geometry. McDougal Littell: Boston, 2007. P. 112. 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. Image taken from: Geometry. McDougal Littell: Boston, 2007. P. 112. Ex. 5 (cont.) STATEMENTS 1. m1 m3 REASON Given 2. mEBA m3 m2 Angle Addition Postulate 3. mEBA m1 m2 Substitution Property of Equality 4. m1 m2 mDBC Angle Additon Postulate 5. mEBA mDBC Transitive Property of Equality 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. Theorem 2.1 Congruence of Segments. Segment congruence is reflexive, symmetric, and transitive. Re flexive: For any segments AB, if AB AB. Symmetric: If AB CD, then CD AB. Transitive: If AB CD and CD EF , then AB EF . Theorem 2.2 Congruence of Angles Angle congruence is reflexive, symmetric, and transitive. Reflexive: For any angle A, mA mA. Symmetric: If A B, then B A. Transitive: If A B and B C , then A C. Ex. 6: Name the property illustrated by the statement. a. If K L and L M , then K M . b. If EF GH , then GH EF . a. Transitive Property of Angle Congruence. b. Symmetric Property of Segment Congruence. 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 Ex. 7 (cont.) STATEMENTS REASONS 1. M is the midpoint of AB. 1. Given 2. AM . MB. 2. Definition of a Midpoint 3. AM MB. 3. Def. of congruent segments 4. AM MB AB. 4. Segment Addition Postulate 5. AM AM AB. 5. Substitution Property of Equality a. 6. 2 AM AB 6. Distributive Property b. 7. AM 1 AB 2 7. Division Property of Equality Turn to page 103 in your book and look at the bottom right example. Refer back to this page for writing proofs! Ex. 8 Go To Chapter 6, Lesson 2, Example 4: http://www.classzone.com/cz/books/geometry_2 007_na/get_chapter_group.htm?cin=2&rg=hel p_with_the_math&at=powerpoint_presentation s&var=powerpoint_presentations Write this example in your notes as Ex. 8.