Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Properties of Equality and Congruence Section 2.6 Objective • Use properties of equality and congruence. Key Vocabulary • • • • Reflexive Property Symmetric Property Transitive Property Logical Reasoning Properties of Equality and Congruence • Just as in algebra where you have the properties of equality that apply to numbers, in geometry we have similar properties called the properties of congruence that apply to geometric figures. – Reflexive Property – Symmetric Property – Transitive Property • Remember – Equal (=) is used with numbers – Congruent (≅) is used with geometric figures Properties of Equality and Congruence Reflexive Property Equality AB AB mA mB Congruence AB AB A A Reflexive Property Jean is the same height as Jean. Properties of Equality and Congruence Symmetric Property Equality If AB CD, then CD AB If mA mB, then mB mA Congruence If AB CD, then CD AB If A B, then B A Symmetric Property If Jean is the same height as Pedro Then Pedro is the same height as Jean Properties of Equality and Congruence Transitive Property Equality If AB CD and CD EF , then AB EF . If mA mB and mB mC , then mA mC. Congruence If AB CD and CD EF , then AB EF . If A B and B C , then A C. Transitive Property If Jean is the same height as Pedro And Pedro is the same Then height as Chris Jean is the same height as Chris Example 1 Name the property that the statement illustrates. a. If GH JK, then JK GH. b. DE = DE c. If P Q and Q R, then P R. SOLUTION a. Symmetric Property of Congruence b. Reflexive Property of Equality c. Transitive Property of Congruence Your Turn: Name the property that the statement illustrates. 1. If DF = FG and FG = GH, then DF = GH. ANSWER Transitive Property of Equality 2. P P ANSWER Reflexive Property of Congruence 3. If mS mT, then mT mS. ANSWER Symmetric Property of Equality Logical Reasoning • In geometry, you are often asked to explain why statements are true. • Logical reasoning is the system that we to explain why something is true. • Logical reasoning is the process of constructing a valid argument from observation and known facts. Example 2 In the diagram, N is the midpoint of MP, and P is the midpoint of NQ. Show that MN = PQ. SOLUTION MN = NP NP = PQ MN = PQ Definition of midpoint Definition of midpoint Transitive Property of Equality Your Turn: 1 and 2 are vertical angles, and 2 3. Show that 1 3. 1 2 ? _____ Theorem 2 3 Given 1 3 _____ ? Property of Congruence ANSWER Vertical Angles; Transitive ALGEBRAIC PROPERTIES OF EQUALITY Addition Property of Equality If a = b, then a + c = b + c. Adding the same number to each side of an equation produces an equivalent equation. Subtraction Property of Equality If a = b, then a – c = b – c. Subtracting the same number to each side of an equation produces an equivalent equation. Multiplication Property of Equality If a = b, then ac = bc. Multiplying each side of an equation by the same nonzero number produces an equivalent equation. Division Property of Equality If a = b, then a/c = b/c. Dividing each side of an equation by the same nonzero number produces an equivalent equation. Substitution Property of Equality If a = b, then you may Substituting a number for a variable replace b with a in any in an equation produces an expression. equivalent equation. WE USE THESE PROPERTIES TO JUSTIFY ALGEBRAIC STEPS AND SOLVE PROBLEMS. THIS IS LOGICAL REASONING Example 3 1 and 2 are both supplementary to 3. Show that 1 2. SOLUTION m1 + m3 = 180° Definition of supplementary angles m2 + m3 = 180° Definition of supplementary angles m1 + m3 = m2 + m3 Substitution Property of Equality m1 = m2 Subtraction Property of Equality 1 2 Definition of congruent angles Your Turn: In the diagram, M is the midpoint of AB. Show that AB = 2 · AM. MB = AM AB = AM + MB AB = AM + AM AB = 2 · AM ANSWER ? Definition of _____ _____ ? Postulate _____ ? Property of Equality Distributive property midpoint; Segment Addition; Substitution Joke Time • What did the fish say when he hit the wall? • DAM!! • What do you call 500 lawyers at the bottom of the sea. • A good start. • You're stranded in a deserted island with Attila the Hun, Adolf Hitler, and a lawyer. You have a revolver with two bullets. What do you do? • Shoot the lawyer twice! Assignment • Section 2.6, pg. 91-94: #1-21 odd, 25, 2935 odd