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

Reflexive Property Jean is the same height as Jean.

Properties of Equality and Congruence Symmetric Property

Symmetric Property Jean is the same height as Pedro Pedro is the same height as Jean If Then

Properties of Equality and Congruence Transitive Property

Transitive Property Jean is the same height as Chris If Jean is the same height as Pedro And Pedro is the same height as Chris Then

Example 1 Name the property that the statement illustrates. a. If GH  JK, then JK  GH. DE = DE b. If P  Q and Q  R, then P  R. c. SOLUTION Symmetric Property of Congruence a. Reflexive Property of Equality b. Transitive Property of Congruence c. 11

Your Turn: Name the property that the statement illustrates. If DF = FG and FG = GH, then DF = GH. 1. ANSWER Transitive Property of Equality P  P 2. ANSWER Reflexive Property of Congruence If mS  mT, then mT  mS. 3. 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 Definition of midpoint Definition of midpoint In the diagram, N is the midpoint of MP, and P is the midpoint of NQ. Show that MN = PQ. SOLUTION MN = NP Definition of midpoint NP = PQ Definition of midpoint MN = PQ Transitive Property of Equality 14

Your Turn: 1 and 2 are vertical angles, and 2  3. Show that 1  3. 1  2 2  3 1  3 Theorem _____ ? Given 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 Definition of supplementary angles m1 + m3 = 180° Definition of supplementary angles m2 + m3 = 180° Substitution Property of Equality m1 + m3 = m2 + m3 Subtraction Property of Equality m1 = m2 Definition of congruent angles 1  2 17

Your Turn: Postulate _____ ? Distributive property Property of Equality Definition of MB = AM AB = AM + MB AB = AM + AM AB = 2 · AM In the diagram, M is the midpoint of AB. Show that AB = 2 · AM. ANSWER 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, 29-35 odd