GEOMETRY UNIT 1 WORKBOOK. CHAPTER 2 Reasoning and Proof

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1 GEOMETRY UNIT 1 WORKBOOK CHAPTER 2 Reasoning and Proof 1

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3 Notes 5 : Using postulates and diagrams, make valid conclusions about points, lines, and planes. I) Reminder: Rules that are accepted without proof are called or. Rules that are proved are called. II) Point, Line, Plane Postulates: a) Write the answers to the questions on the Partner Investigation below. 1) 2) 3) 4) 5) 6) 7) Hint: Look at your answers from these problems above. b) Use your observations from the Partner Investigation to complete the following. Through any two points you can only draw line(s). (#1) To draw a line, you must have at least points. (#2) If two lines intersect, then their intersection is exactly point(s). (#4) You can draw exactly plane(s) through any three noncollinear points. (#5) To make a plane, you need at least noncollinear point(s). (#3) If two points lie in a plane, then the line going through them also lies in the. If two planes intersect, then their intersection is a. (#7) 3

4 Write a proof in the following example. 1) GIVEN: A B, B C PROVE: A C 4

5 Take notes on pg. 136 in textbooks (key concept) & fill in the blanks below Example 1 provides an example of an algebraic proof. Write a proof for Example Show each step on a different line. 1) 2(5 3a) 4(a + 7) = 92 2) -4(11x + 2) = 80 In your own words, what does simplify/combine like terms mean? 5

6 Take notes on pg in textbook on Segment Addition Postulate & Example 1 Draw a picture Fill in the blanks Definition of congruent segments When do you think definition of congruent segments is used? What is always the first reason? What is always the last statement? 6

7 Warm-Up Given: WY = YZ YZ XZ XZ WX Prove: WY WX WY = YZ WW YY YY XX 4. WW XX 5. XX WW Quick Notes Definition of Midpoint:. Given: AB DE B is the midpoint of AC. E is the midpoint of DF. Prove: BC EF AB = DE Given Given Definition of Midpoint Given Definition of Midpoint 7. DE = BC BC = EF

8 In-Class Practice Complete each proof. Given: BC = DE Prove: AB + DE = AC Proof: BC = DE Segment Addition Postulate AB + DE = AC Given: X is the midpoint of MN and MX = RX Prove: XN RX X is the midpoint of MN XN = MX MX = RX 4. XN = RX 5. XN RX Given: GD BC BC FH FH AE Prove: AE GD Given Given Transitive Property 4. Given 5. Transitive 6. You can now fill in # s 1 7 on proof reasons sheet!! 8

9 Take notes on the following topics beginning on page 151 of your textbook. Draw pictures to help. *Angle Addition Postulate: Supplement Theorem: *What does supplementary mean? Complement Theorem: *What does complementary mean? Congruent Supplements Theorem: Congruent Complements Theorem: *Vertical Angles Theorem: *Indicates reasons that are often used in proofs 9

10 Read through this section, highlight/underline important pieces, and write questions you have. When thinking about definition of congruent segments, when do you think definition of congruent angles is used? This is not explicitly in this section. Just think about it using your reasoning skills. 10

11 Write a proof for the following: Given: 3 2 Prove: Warm-Up Quick Notes Given: 1 and 4 form a linear pair, and m 3 + m 1 = 180. Prove: 3 and 4 are congruent. 1 and 4 are supplementary m 3 + m 1 = m 3 and m 1 are supplementary In-Class Practice For numbers 1 & 2, find the value of x and name the theorems that justify your work. m 1 = (x + 10) m 6 = (7x 24) m 2 = (3x + 18) m 7 = (5x + 14) x = x = Theorem: Theorem: 11

12 This image cannot currently be displayed. Write a proof for the following: Given: m 4=120 o, 2 5, 5 4 Prove: m 2=120 o Reason 1) m 4=120 o 2) 2 5; 5 4 3) 2 4 4) m 2 = m 4 4. Given: 6 5 Prove: Write a proof for the following: Given: 1 5; 4 and 5 are a linear pair Prove: 1 is supplementary to m 1 = m 5 4 and 5 are a linear pair 4. 4 and 5 are supplementary 5. m 4 + m 5 = 180 o 6. m 4 + m 1 = 180 o is supplementary to 4 You can now fill in # s on proof reasons sheet!! 12

13 Geometry REVIEW 5 8 Name For numbers 1 3, determine if the statement is always (A), sometimes (S), or never (N) true. If two points lie in a plane, then the entire line containing those points lies in that plane. Two lines intersect to form right angles. Three noncollinear points are contained in Plane Y. For numbers 4 6, find the measure of the indicated angle and name the theorems that justify your work. 4. If m 1 = (x + 50) and m 2 = (3x 20), find m m 1 = Theorem: 5. If m 1 is twice m 2, find m m 1 = Theorem: 6. If ABC EFG and m ABC = 41, find m GFH. m GFH = Theorem: For numbers 7 12, state the definition, property, postulate, or theorem that justifies each statement. 7. If X is the midpoint of CD, then CX = XD. 8. If K P and P T, then K T. 9. If m W + m H = 90 and m H = 20, then m W + 20 = If A and B are complementary and A and D are complementary, then B D. 1 If AT DR then AT = DR. 1 AB + BC = AC 13

14 1 Complete the proof by supplying the missing information If 2x 7 = 4, then x = 11 2 Proof 14. Given: M is the midpoint of AB MB BX Prove: AM BX A M B X Reason Bank: Addition Property Congruent complements theorem Congruent supplements theorem Definition of complementary angles Definition of congruent angles Definition of congruent segments Definition of midpoint Definition of supplementary angles Distributive Property Division Property Reflexive Property Midpoint Theorem Multiplication Property Segment Addition Postulate Substitution Property Subtraction Property Supplement Theorem Symmetric Property Transitive Property Vertical angles are Congruent M is the midpoint of AB AM = MB Given Given: Ð1 Ð2 Ð2 Ð3 Ð3 Ð4 Prove: Ð1 Ð Ð1 Ð2 Ð2 Ð3 Ð1 Ð3 4. Ð3 Ð4 5. Ð1 Ð