Now we are going to look at Geometry Proofs
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Geometry 2.5/2.6 Introduction to Proofs Name: _________________________ Last class, we worked on writing logical statements and making arguments on whether or not those statements were true or false. Today we are going to use properties from algebra, to prove different statements. Let’s start with a little review… Properties of Equality from Algebra Let a, b and c be real numbers. Addition Property: If π = π, then _____________________________. Subtraction Property: If π = π , then __________________________. Multiplication Property: If π = π then _______________________. Division Property: If π = π and c οΉ 0 then ______________________. Distributive Property: π(π + π) = ______________________ *Combining Like Terms is part of the Distributive Property.* Substitution Property: If a = b then either a or b may be substituted for the other in any equation (or inequality). Properties of Equality from Geometry Properties of Congruence from Geometry Reflexive Property: π = ____________ Reflexive Property: π ο ______________ Symmetric Property: If π = π, then ___________. Symmetric Property: If π ο π, then ____________. Transitive Property: Transitive Property: If π = π and π = π then ________. If π ο π and π ο π then __________. Let’s try some examples of Algebra Proofs. Example 1: Statement 8π₯ – 7 = 12π₯ + 9 − 7 = 4π₯ + 9 −16 = 4π₯ −4 = π₯ π₯ = −4 Reason Given Example 2 Statement Reason 3( 2π₯ – 4) + 4π₯ = 18 6π₯ – 12 + 4π₯ = 18 10π₯ – 12 = 18 10π₯ = 30 π₯ = 3 Example 3: Statement 4(5π₯ − 9) = −2(π₯ + 7) Reason Given Now we are going to look at Geometry Proofs: Proof – a logical argument that shows a statement is TRUE. Two – Column Proof : numbered _____________________ and corresponding __________________ that show an argument in a logical order. Valid Reasons for a Proof: When we write proofs, we always write the _____________________ information first. The last statement in a proof should always be ___________________________________. ο· ο· ο· ο· Postulates are rules that are accepted without proof. Rules that are proved are called theorems. Postulates and theorems are often written in conditional form (if- then). Unlike the converse of a definition, the converse of a postulate or theorem cannot be assumed to be true! Let’s see if we can try some examples. Directions: Draw a conclusion based on the diagram or given information. 1. Given: π ππ π‘βπ ππππππππ‘ ππ Μ Μ Μ Μ ππ π Statement: ________________________________ π π Reason: ________________________________ 2. Given: ββββββ π΅π· πππ πππ‘π β‘π΄π΅πΆ π΅ π· πΆ Statement: ________________________________ Reason: ________________________________ 3. 1 2 Statement: ________________________________ Reason: ________________________________ 4. Given: β ⊥ π. β 1 π Statement: ________________________________ Reason: ________________________________ 5. Given: β‘1 πππ β‘2 πππ π π’ππππππππ‘πππ¦. Statement: ________________________________ Reason: ________________________________ 6. Make a conclusion about β‘π΄π΅π· πππ β‘π·π΅πΆ. π΄ π· π΅ πΆ Statement: ________________________________ Reason: ________________________________ Μ Μ Μ Μ . 7. Give a statement about π΄π· π΄ π΅ π· πΆ Statement: ________________________________ Reason: ________________________________ Complete the 2-column proof for each of the following: 1. Given: π΅ ππ π‘βπ ππππππππ‘ ππ Μ Μ Μ Μ π΄πΆ . Prove: π΄π΅ + πΆπ· = π΅π· A C B STATEMENTS 1. π΅ ππ π‘βπ ππππππππ‘ ππ Μ Μ Μ Μ π΄πΆ . 1. Μ Μ Μ Μ 2. Μ Μ Μ Μ π΄π΅ ≅ π΅πΆ 2. 3. 3. Definition of Congruence 4. π΅πΆ + πΆπ· = ____________ 4. Segment Addition Postulate 5. 5. 2. π΄π΅ + πΆπ· = π΅π· D REASONS Given: ∠1 πππ ∠2 πππ π π‘ππππβπ‘ ππππππ Prove: ∠ 1 ≅ ∠2 1 2 STATEMENTS 1. ∠1 πππ ∠2 πππ π π‘ππππβπ‘ ππππππ REASONS 1. 2. π∠1 = 180° ; π∠2 = 180° 2. 3. 3. Transitive Property of Equality 4. ∠ 1 ≅ ∠2 4.
