GEOMETRY. 2.5 Proving Statements about Segments and Angles

Transcription

1 GEOMETRY 2.5 Proving Statements about Segments and Angles

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3 ESSENTIAL QUESTION How can I prove a geometric statement?

4 REVIEW! Today we are starting proofs. This means we will be using ALL of the theorems and postulates you have learned this year. Let s review.

5 REVIEW: ANGLE ADDITION POSTULATE A B D C If B is in the interior of ADC, then m ADB + m BDC = m ADC

6 REVIEW: SEGMENT ADDITION POSTULATE If B is between A and C, then AB + BC = AC. If AB + BC = AC, then B is between A and C. A B C AB AC BC

7 REVIEW: DEF. OF CONGRUENT SEGMENTS Two segments are congruent if and only if they have the same length. This is a biconditional: 1) If two segments are congruent, then they have the same length. 2) If two segments have the same length, then they are congruent. September 19, 2016 GEOMETRY 2.5 PROVING STATEMENTS ABOUT SEGMENTS AND ANGLES

8 IN SYMBOLS: If AB CD, then AB = CD. If RS = TV, then RS TV. (Don t forget this ) September 19, 2016 GEOMETRY 2.5 PROVING STATEMENTS ABOUT SEGMENTS AND ANGLES

9 WRITING A TWO-COLUMN PROOF We use deductive reasoning: Definitions, properties, postulates, and theorems One of the formats for a proof is a two-column proof. Statements Reasons

10 EXAMPLE 1 What is the measure of the entire angle?

11 EXAMPLE 2 M N P If MN = 10, and MP = 24.5, find NP. Solution By SAP, MN + NP = MP so 10 + NP = 24.5 and NP = 14.5

12 EXAMPLE 3 m 1 = m 3 m 1 + m 2 m CBD m EBA = m CBD

13 YOUR TURN Seg. Add. Prop. Trans. Prop. of Equality Subtr. Prop. of Equality

14 EXAMPLE 4 Write a two-column proof. Given: Prove: Statements Reasons 1. Given D 2. Angle Addition Postulate 3. Substitution 4. Angle Addition Postulate 5. Transitive Property E

15 REMEMBER THESE FROM 2.4? Algebraic Properties of Equality Geometric Properties of Congruence Real Numbers Segments Angles Reflexive a = a AB AB A A Symmetric If a = b, then b = a Transitive If a = b, and b = c, then a = c If AB CD, then CD AB If AB CD, and CD EF, then AB EF If A B, then B A If A B, and B C, then A C 2.4 ALGEBRAIC REASONING

16 THEOREM 2.1 Properties of Segment Congruence. Segment congruence is reflexive, symmetric, and transitive. Reflexive: Symmetric: Transitive: AB AB If AB CD, then CD AB If AB CD, and CD RS, then AB RS Remember: a THEOREM is a statement that is proven to be true. September 19, 2016 GEOMETRY 2.5 PROVING STATEMENTS ABOUT SEGMENTS AND ANGLES

17 THEOREM 2.2 Properties of Angle Congruence. Angle congruence is reflexive, symmetric and transitive. Reflexive: Symmetric: Transitive: ABC ABC If A B, then B A If A B, and B C, then A C The proofs are similar to those for segment congruence and will not be given here. GEOMETRY 2.6 PROVING STATEMENTS ABOUT ANGLES 17

18 EXAMPLE 6

19 Food for Thought: There is no magical way to learn to do proofs. Doing proofs requires hard thinking, serious effort, memorization, a lot of writing, and dedication. There are no shortcuts, there are no quick easy answers. To be successful at proof, you must know every definition, postulate and theorem. Looking them up in a book is no substitute. Every year, millions of students across the country learn proofs. You can do it, too! September 19, 2016 GEOMETRY 2.5 PROVING STATEMENTS ABOUT SEGMENTS AND ANGLES

20 PROOF: SYMMETRIC PROPERTY Given: AB CD. Prove: CD AB. Statements Reasons 1. AB CD 1. Given 2. AB = CD 2. We Def. just seg. had this. 3. CD = AB 3. Symm. Prop. 4. CD AB 4. Def. seg. Latin: quod erat demonstrandum That which was to be demonstrated. Step 3, although seemingly trivial and unnecessary, is important: we need it to show that segment congruence is symmetric just as in algebra. September 19, 2016 GEOMETRY 2.5 PROVING STATEMENTS ABOUT SEGMENTS AND ANGLES

21 IS ALL THIS NECESSARY? September 19, 2016 GEOMETRY 2.5 PROVING STATEMENTS ABOUT SEGMENTS AND ANGLES

22 EXAMPLE 7 Given AB = 20, M is the midpoint of AB. Prove: AM = 10. A M B Statements Reasons 1. AB = Given 2. M is midpt of AB 2. Given 3. AM MB 3. Def. of midpoint 4. AM = MB 4. Def. of congruent seg. 5. AM + MB = AB 5. Seg. Add. Post. (SAP) 6. AM + AM = Substitution (4,5 & 1,5) 7. 2AM = Simplify 8. AM = Division Property September 19, 2016 QED GEOMETRY 2.5 PROVING STATEMENTS ABOUT SEGMENTS AND ANGLES

23 EXAMPLE 8 Given: AB CD, B is the midpoint of AC. Prove: BC CD Statements Reasons 1. AB CD 2. B is the midpoint of AC 3. AB BC 4. BC AB 1. Given 2. Given 3. Def. of Midpoint 4. Sym. Prop. of Seg. 5. BC CD 5. Trans. Prop. Of Seg.

24 EXAMPLE 9: USING ALGEBRA Solve for x. AC = x + 8 6x + 12 A B C Statements Reasons 1. AC = Given 2. AB = 3x + 8, BC = 6x Given 3. AB + BC = AC 3. Seg. Add. Post. (SAP) 4. (3x + 8) + (6x + 12) = Substitution (2,3 & 1,3) 5. 9x + 20 = Simplify 6. 9x = Subtraction Property 7. x = Division Property September 19, 2016 QED GEOMETRY 2.5 PROVING STATEMENTS ABOUT SEGMENTS AND ANGLES

25 ASSIGNMENT September 19, 2016 GEOMETRY 2.5 PROVING STATEMENTS ABOUT SEGMENTS AND ANGLES