Notes #35: Inequalities and Reasoning (Sections 6.1 and 6.2)

Transcription

1 Geometry Rules! Chapter 6 Notes Notes #35: Inequalities and Reasoning (Sections 6.1 and 6.) lgebraic Inequalities in if/then form: (Hint: given that the if statement is true, can we say that the then statement must be true?) If it is not true, provide a counterexample (explain why). 1.) If c 3 < 5, then c < 8..) If d 3 > 4, then d > 5. 3.) If x > 4, then x > 3. 4.) If x > 3, then x > 4. Inequalities in Geometry: Whole > Part 5.) Fill in the blank with <, >, or = Y a) Y + Y b) Y c) Y 6.) Can you conclude each statement? (yes/no) X a) m OX = ½ m O b) m OX + m XO = m O c) m O > m XO d) m O < 90 e) m XO > m OX f) m O > m OX O Inverses/Contrapositives: Original: If Mrs. Spragg has a hermit crab, then she has a pet. If p ( ), then q ( ). Contrapositive: If not q ( ), then not p ( ).

2 Converse: - - If q, then p. Inverse: If not p, then not q. Write the contrapositive, converse, and inverse of each. State whether each is true or false: (T/F) 7.) Original: If I live in Encinitas, then I live in California. Contrapositive: (If not q, then not p.) Converse: (If q, then p) Inverse: (If not p, then not q.) statement and its are logically equivalent. statement s converse and its are logically equivalent. Deductive Reasoning: Consider the given statements as true and write it in if, then form. What can you conclude from each additional piece of information? (It might help to write its logically equivalent contrapositive, too.) 8.) ll US Presidents are at least 35 years old. Original: If, then. Contrapositive: If, then. a) Tracy Smith is 38 years old. b) George ush is the president.

3 c) Mr. Rich is not the president. d) Mr. King is years old ) ll runners are athletes. Original: If, then. Contrapositive: If, then. a) Leroy is a runner b) Pam is not an athlete c) Sara is an athlete d) Jim is not a runner lgebra Practice: Simplify (factor first, when possible!) 10.) 1x 18x ) 4ab 16ab ) c d 4c+ 4d 13.) 6x 8y ) x x + x x ) x + 4x x+ x

4 Notes #36: Indirect Proof, Proofs by Contradiction (Section 6.3) Indirect proofs are used to prove statements that are difficult or impossible to prove using accepted theorems, postulates, and definitions. Indirect proof, or proof by contradiction, proves a statement by demonstrating that its (demonstrating that the contrapositive is true). 1.) Given: Scott hit a single and a home run on Friday. Prove: Scott played in a baseball game on Friday. ssume temporarily that Scott did not. Then he could not have. However, this contradicts the given that. Therefore, our assumption was wrong, and. Recipe for an indirect proof: ssume temporarily (the opposite of the conclusion) (Make logical conclusions until you reach a problem) However, this contradicts the given because (explain the problem) Therefore, my assumption is false and (the original conclusion must be true) Write the first sentence of an indirect proof for each of the following statements:.) If m<x = 60, then m< = ) If m = n, then n p.

5 Write an indirect proof for each: ) Given: diagram Prove: X isn t a median of C 5 X 8 C ssume temporarily that. Then, X would be the, and by, =. However, this contradicts the given because Therefore, my assumption is false, and. 5.) Given: C, m< = 110 Prove: < is not a right angle 6.) Given: Jon enters the house with a dry umbrella. Prove: It is not raining outside.

6 lgebra Practice: Simplify (factor first, when possible!) ) 8x + 10x+ 3 8x + x 3 8.) x 4 7x x ) 6x 15x 9 4x 6x 4 10.) 5x 10x x 10x 11.) 3 x x x x 5x 4 1.) (-m 3 n 5 )(5mn ) 13.) (4x 3 y ) 14.) (-j 4 k ) 3 (5jk 3 ) 15.) (x + 4)

7 Notes #37: Triangle Inequalities and Ratios (Sections 6.4 and 7.1) The largest angle of a triangle is opposite the side of the triangle. The smallest angle of a triangle is opposite the side of the triangle. Name the largest and smallest angle of the triangle: (not necessarily drawn to scale) 1.).) a + 1 X Y C Largest: Smallest: a - 1 Z a Largest: Smallest: Name the longest and shortest side of the triangle: (not necessarily drawn to scale) 3.) 4.) 46 4 C Longest: Shortest: Z Shortest: Name the largest and smallest angle: (not necessarily drawn to scale) 5.) X 63 7 Y Longest: Largest: Smallest: Name the longest side: (not necessarily drawn to scale) 6.) S 48 Longest: R 55 T 5 65 U

8 The Triangle Inequality The sum of the lengths of of a triangle is greater than the length of the. When given three side lengths of a possible triangle, they can form a triangle if: (short side) + (short side) > (long side) Is it possible for a triangle to have sides with length: (Yes/No) 7) 8, 7, 6 8) 9, 9, 19 9), 3, 4 When given two side lengths of a triangle, the third side of the triangle must be between their difference and their sum. (length length) < third side < (length + length) The lengths of two sides of a triangle are given. The length of the third side must be greater than but less than. 10.) 3, 7 11.) 1, 19 1.) 3a, 4a +

9 lgebra Practice: Simplify (factor first, if possible!) ) (3a b 3 ) (-5a 3 b 6 ) 14.) (4d 5) 15.) (x y 3 ) + (3xy 4 )(5x 3 y ) 16.) 36 4x 6x + x i 17.) 9x 4x 9 8xy x + 3x 4x 1 + 3x x x x x Ratios: Comparing numbers with division 18.) Find each ratio: a) D:DC 8 C 6 b) m<:m<d 40 D c) D: Perimeter of CD 19.) If x = 5, y = 10, z = 3 find each ratio: a) x to y b) (x + z) to y c) x+ y 7z

10 Ratio word problems: - attach an x to each part of the ratio - write and solve an equation using these expressions ) The ratio of two complementary angles is 1:. Find each angle. 1.) The ratio of two supplementary angles is 7:. Find each angle.) The ratio of the angles of a triangle is :3:4. Find each angle. 3.) The ratio of the angles of a pentagon is :3:4:4:5. Find each angle.

11 Classwork #38: Chapter 6 Review Classify each if-then statement as True or False. 1. If x > 6, then x > 5.. If x> 1, then x> 13. Complete each statement by writing <, =, or >. 3. X X 4. C 5. C C 1 D 6. m 1 70 º 7. m 70º F E State whether each statement is True or False. Then write the (a) contrapositive, (b) converse, and (c) inverse and state whether each is True or False. T/F 8. If two lines do not intersect, then they are skew. (a) (contrapositive) (b) (converse) (c) (inverse) Complete the indirect proof: 9. Given: Quadrilateral CD, m = 80. Prove: CD is not a rectangle. ssume temporarily that. Then m = because. However, this contradicts the that. Therefore, was wrong and. 10. Is it possible for a triangle to have sides with the given lengths: (yes/no) (a), 4, 6 (b) 5, 5, The lengths of two sides of a triangle are given. The length of the third side must be greater than but less than. (a), 8 (b) b, b + 3

12 Name the longest and shortest side of the given figure: 13. Name the largest and the smallest side of the given triangle: O G 4x - 10 D S x + 10 x The angles of a triangle are in the ratio of :3:4. Find the angles. Simplify: 15. (3c 3 d 4 ) (-cd 5 ) 16. (4x 3) a + a 9a a a 18. m + 5m+ 3 m y 10y y+ 6 4y 36 15y 3 i 0. w w w w 1 6w w 1

13 HW #38: Study Guide (Sections , 7.1) Classify each if-then statement as True or False. If false, provide a counterexample. 1. If x > 6, then x > 3.. If x > 3, then x >. 3. If x> 15, then x> 17. Complete each statement by writing <, =, or >. X 4. X 5. C C 1 D 6. X C 7. m 1 75 º F E State whether each statement is True or False. Then write the (a) contrapositive, (b) converse, and (c) inverse. State whether each is True or False. T/F 8. If two lines do not intersect, then they are skew. (a) (b) (c) ccept the given statement as true. What can you conclude by using the given statement together with each additional statement? 9. Given: ll students love math. (a) randen is a student. (c) lan loves math. (b) Kim is not a student. (d) Jim does not love math. What is the first sentence of an indirect proof of the statement shown? 10. If = C, then DE = EF. 11. If m = 30, then m 30. Complete the indirect proof: 1. Given: Quadrilateral CD, m = 80. Prove: CD is not a square. ssume temporarily that. Then, m = because. However, this contradicts the because Therefore, was wrong and.

14 Is it possible for a triangle to have sides with the indicated lengths? (Yes/No) 13. 1,, , 4, The lengths of two sides of a triangle are given. The length of the third sides must be greater than but less than , a, a + 6 If the diagram was drawn to scale, name the largest and smallest angles. 17. In C, = 11, C = 1, and C = b R b Y 10 8 Z Largest: Smallest: Q b - 1 P Largest: Smallest: 1 3 Largest: Smallest: If the diagram was drawn to scale, name the shortest and longest sides. 0. In C, m< = 60, m< = 59, m<c = U x Longest: Shortest: Simplify: 6ab 3. 1ab x - T S Longest: Shortest: D 59 C Longest: Shortest: 4. x+ y for x= 3, y =, z = 1 z x Write and equation and solve: 5. The ratio of the angles of a triangle is 1:3:5. Find the angles. 6. The ratio of two complementary angles is 5:1. Find the angles. 7. The ratio of the angles of a pentagon is 6: 8: 9: 11: 11. Find the angles.

15 Simplify: (check that factoring!) ) (5m 3 n )(-4mn 8 ) 9.) (-3a 5 b ) 3 (ab 4 ) 30.) (6x 3 y ) (-3x 5 y ) 3 31.) 3a(a 3b) 3.) (x 1) 3 33.) x 5x+ 6 x 9 34.) x x 1 x 35.) 4mn p 8 36mnp ) 4x + x 3 6x 6 37.) m 6 m 3 4m+ 8 m ) 3 y + y 4y 8 y 1 i y + 3y+ 3y+ y 39.) 4c 1 c + 5c 3 6c 6 3c + 9c 3 40.) x x x x i x x x x