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lesson 2.6 Geometry.notebook October 16, 2015 Warm Up~ Identify the property being used. 1. 5x = 5x __________________ 2. AB AB _________________ 3. If y=2x, then 2x=y. _________ 4. ___________ 5. If CD GH, then GH CD ______________ 6. Solve 5(x 12) + 3x = 9x 21. Justify each step. Statements Reasons lesson 2.6 Geometry.notebook October 16, 2015 lesson 2.6 Geometry.notebook October 16, 2015 lesson 2.6 Geometry.notebook October 16, 2015 lesson 2.6 Geometry.notebook October 16, 2015 Warm Up Algebraic Proof: Write a Reason for each step Given: B is the midpoint of AC Prove: y=5 A B Statement 1) " " Reason 1) 2) 2) 3) 3) 4) 4) 5) 5) 6) 6) C lesson 2.6 Geometry.notebook October 16, 2015 lesson 2.6 Geometry.notebook October 16, 2015 lesson 2.6 Geometry.notebook October 16, 2015 lesson 2.6 Geometry.notebook October 16, 2015 lesson 2.6 Geometry.notebook October 16, 2015 lesson 2.6 Geometry.notebook October 16, 2015 lesson 2.6 Geometry.notebook October 16, 2015 2.6 Geometric Proof GOALS~ 1) Write twocolumn proofs. 2) Prove geometric theorems by using deductive reasoning. lesson 2.6 Geometry.notebook October 16, 2015 Definitions~2 total 1) theorem 2) twocolumn proof Theorems~7 total 1) Linear Pair Theorem 2) Congruent Supplements Theorem 3) Congruent Complements Theorem 4) Right Angle Congruence Theorem 5) Vertical Angles Theorem 6) Congruent Supplements are Rt. Ang. 7) Common Segments Theorem lesson 2.6 Geometry.notebook October 16, 2015 2.6 Geometric Proof Definition: Theorem A theorem is any statement that you can prove. Once you have proven a theorem, you can use it as a reason in later proofs. lesson 2.6 Geometry.notebook October 16, 2015 When writing a proof, it is important to justify each logical step with a reason. You can use symbols and abbreviations, but they must be clear enough so that anyone who reads your proof will understand them. Given Prove lesson 2.6 Geometry.notebook Example 1: Writing Justifications Write a justification for each step Given: ∠A and ∠B are supplementary and m∠A = 45°. Prove: m∠B=135° October 16, 2015 lesson 2.6 Geometry.notebook October 16, 2015 With a Partner Write a justification for each step, given that B is the midpoint of AC and AB ≅ EF. Given: B is the midpoint of AC and AB ≅ EF Prove: Statement Reason lesson 2.6 Geometry.notebook October 16, 2015 1. Linear Pair Theorem they are supplementary If two angles form a linear pair, then _____________________. ÐA ÐB lesson 2.6 Geometry.notebook October 16, 2015 2. Congruent Supplements Theorem If two angles are supplementary to the same angle (or to two the two angles are congruent congruent angles), then _____________________________. mÐ1 = 80 mÐ2 = 100 mÐ3 = 80 lesson 2.6 Geometry.notebook October 16, 2015 3. Right Angle Congruence Theorem congruent All right angles are ___________. 4. Congruent Complements Theorem If two angles are complementary to the same angle (or to two then the two angles are congruent congruent angles), then __________________________________. mÐ1 = 40 mÐ2 = 50 mÐ3 = 40 Proof of Congruent Complements Theorem lesson 2.6 Geometry.notebook 5. Vertical Angles Theorem: congruent Vertical angles are ___________ October 16, 2015 lesson 2.6 Geometry.notebook October 16, 2015 6. Congruent Supplements are Right Angles If two congruent angles are supplementary, then each angle is ______________ lesson 2.6 Geometry.notebook October 16, 2015 7. Common Segments Theorem Given collinear points A, B, C, D arranged as shown, ~ If AB = CD, then ________ A B C D lesson 2.6 Geometry.notebook October 16, 2015 A geometric proof begins with Given and Prove statements, which restate the hypothesis and conclusion of the conjecture. In a twocolumn proof, you list the steps of the proof in the left column. You write the matching reason for each step in the right column. lesson 2.6 Geometry.notebook October 16, 2015 lesson 2.6 Geometry.notebook October 16, 2015 Fill in the blanks to complete a twocolumn proof of one case of the Congruent Supplements Theorem. Given: ∠1 and ∠2 are supplementary, and ∠2 and ∠3 are supplementary. Prove: ∠1 ≅ ∠3 a) b) c) d) lesson 2.6 Geometry.notebook October 16, 2015 Before you start writing a proof, you should plan out your logic. Sometimes you will be given a plan for a more challenging proof. This plan will detail the major steps of the proof for you. lesson 2.6 Geometry.notebook October 16, 2015 Flow Game https://games.cdn.famobi.com/html5games/f/flowfree/v4/? fg_domain=play.famobi.com&fg_aid=A10001&fg_uid=8557fc8f26b34bc1a770 d4fa798a30ca&fg_pid=4638e3204444451481c4d80a8c662371&fg_beat=191 lesson 2.6 Geometry.notebook October 16, 2015 lesson 2.6 Geometry.notebook October 16, 2015 Example 3: Writing a TwoColumn Proof from a Plan lesson 2.6 Geometry.notebook October 16, 2015 Your Turn Write a justification for each step given: m∠ABC = 90° and m∠1 = 4(m∠2). prove: m∠2 = 18° Statement 1. m∠ABC = 90° and m∠1 = 4m∠2 2. m∠1 + m∠2 = m∠ABC 3. 4m∠2 + m∠2 = 90° 4. 5m∠2 = 90° 5. m∠2 = 18° Reason 1. Given 2. ∠ Add. Post. 3. Subst. 4. Simplify 5. Div. Prop. of =. lesson 2.6 Geometry.notebook Homework: Due Next Class 1) 2.6 pg 113 #14, 68 2) Worksheet: "Angle Proof" October 16, 2015 lesson 2.6 Geometry.notebook October 16, 2015
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