Name: Period: 2.1 Inductive Reasoning What s Next? Example in class: 1, 1, 2, 3, 5, 8,, What are the next two terms in each sequence? 1) 1, 10, 100, 1000,, 2) 7, 3, -1, -5, -9, -13,, 3) 1, 3, 6, 10, 15, 21,, 4) 32, 30, 26, 20, 12, 2,, 5) 1, 2, 4, 8, 16, 32,, 6) 1, 1, 1, 2, 6 3 2 3 Use inductive reasoning to draw the next shape in each picture 7) 8) 9)
10) What is the next shape? 11) Which one of the numbers does not belong in the following series? Explain why. 3, 4, 8, 9, 10, 18, 19, 38 12) The price of eyeglass frames is based on a pattern. How much is the one labeled???? $221 $111 $212 $122??? 13) Remove 4 matchsticks so that you have 5 squares left. Once you find one solution, try to find another way to do the puzzle.
Name: Period: 2.2 Algebraic Proof Solve each equation. Write a justification for each step. The first one is done for you Example: 1) -5 = 3c + 1 1) Given 2) -1-1 2) Subtraction Property of equality 3) -6 = 3c 3) Simplify 4) 3 3 4) Division Property of equality 5) -2 = c 5) Simplify 6) c = -2 6) Symmetric Property of Equality Take as many steps as you need to be complete and justify every step! 1) 2p 30 = -4p + 6 2) z 3 2 = 10
3) - (w + 3) = 72 4) ½ (b 16) = 13 5) Finish the proof to solve for n. Then find the length of RT. Write a justification for each step. Include as many steps as you need. 1) RS + ST = RT 1) Segment Addition Postulate 2) 5n + 30 = 9n 5 2) 3) 30 = 4n 5 3) 4) 4) Addition property of equality 5) = n 5) 6) n = 6) 7) RT = 7)
Name: Period: 2.3 Deductive Reasoning In problems 1 & 2, complete an algebraic proof to determine the solution of each equation. Please list each step you use, and justify each step with a reason. 1) 3(x + 2) 1 = 4(x + 6) + 19 2) 3 = m 7 5 3 m 3) ABC is equilateral. Is ABD equilateral? Explain your answer with deductive reasoning.
4) I found this picture on the back of a package of goldfish crackers. Critique the reasoning. 5.) Find counter-examples to the following statements. a) All prime numbers are odd. b) All even numbers have a 2, 4, 6, or 8 in the ones digit place. c) All mammals have fur. d) If you are in Geometry at TVHS, then you are in Mrs. Dasse s classroom right now. 6.) Use the definition: A number is even if it is divisible by 2 to explain why 0 is an even number. 7.) To be eligible to hold the office of the president of the United States, a candidate must be 35 years old, be a natural born citizen, and must have been a U.S. resident for at least 14 years. Given this information what conclusions can you draw from the following scenario? Michael is not eligible to be the president of the United States even though Michael has lived in the United States for 16 years.
Name: Period: 2.4 Deductive Reasoning Four couples decided to go camping to the state forest one weekend. Each couple traveled in a different van and each chose a separate camping spot. The camping sites were all labeled with a space number and while they were in the same area, the sites were not touching. Using the clues and the grid below, determine the full name of each couple, the color of their van, and the number of their camping space. 1. Bill, who is not married to Laura, didn't drive a black van. 2. Chuck and his wife Brenda were not camped in space #35. Brenda's last name is not Forrest. 3. The Lewis couple, who drove a tan van, camped in space #31. 4. Tom camped in a space numbered lower than the one Cindy camped in but higher than the couple who drove in the red van did. 5. Tom isn't married to Mary Tread. Steve Branch didn't drive a blue van. 6. The couple driving the black van camped in space #43.
Fill in the blank squares so that each row and each column contain all of the digits 1 thru 4 (or whatever the size of the puzzle is). The heavy lines indicate areas (called cages) that contain groups of numbers that can be combined (in any order) to produce the result shown in the cage, with the indicated math operation. For example, 8 means you can multiply the values together to produce 8. Numbers in cages may repeat, as long as they are not in the same row or column. Numbers in cages may repeat, as long as they are not in the same row or column.
Name: Period: 2.5 Geometric Proof Geometric Proof Take as many steps as you need to prove each statement. Provide a reason for each step 1. Given AB HY Prove AB = HY 2. Given m A = 30 and m B = m A Prove m B = 30. C... D A B 3. Given 1 and 2 form a linear pair Prove: 1 and 2 are supplementary 1) 1 and 2 form a linear pair 1) 2) AD and AB form a line 2) Definition of 3) m DAB = 3) Definition of 4) 4) Angle Addition Postulate 5) 5) Substitution (steps 3 & 4) 6) 1 and 2 are supplementary 6)
1 2 3 4. Given: JKL is a right angle, 2 3 Prove: 1 and 3 are complementary 1) JKL is a right angle 1) 2) m JKL = 90 0 2) 3) 3) Angle Addition Postulate 4) m 1 + m 2 = 90 0 4) 5) 2 3 5) 6) 6) Def. of s 7) m 1 + m 3 = 90 0 7) 8) 8) Def. of 5. Given CD EF and CD FG Prove: F is the midpoint of EG 1) 1) 2) 2) 3) 3) Transitive 4) 4)
Name: Period: 2.6 Angle Pairs Angle Pair-adise 1) Supplementary Angles: Two angles whose measures sum to 180 o. Use the diagram above to prove that 2 angles which are each supplementary to a 3 rd angle must be congruent. 2) Vertical Angles: Two angles whose sides form two pairs of opposite rays Use the diagram at the right to prove that vertical angles are congruent. 3) Right angle: An angle whose measure is 90 0. (aka Mr. S s Favorite theorem ) 1 Prove: All right angles are congruent. 2
4) Given: ABC is a right angle, find the measures of ABD and DBC 5) What type of angles are 1 and 2? Given: the m 1 = 2x + 3 and m 2 = 3x + 2 Find the m 3 6) What kind of angles are ABE and EBC? Given: m ABE = 2x + 5 and m EBC = x + 4 Find the m DBC 7) What kind of angles are DBC and ABD? Given: m ABD = 4x + 5 and m DBC = 2x + 1 Find m CBE 8) What kind of angles are 4 and 2? Given: m 4 = 5y 9 and m 2 = 2y + 3 Find m 3
Name: Period: 2.7 More Proofs Prove each statement: 1. Given that AB CD and CD, EF prove that AB EF Statements Reasons 2. Use the diagram to prove that LK JL Statements Reasons 3. Given that 3 = 40, 1 2, 2 3. Prove that m 1 = 40 Statements Reasons
4. Use the vertical angles theorem to prove that when 2 lines are perpendicular to one another, that all angles measure 90 Statements Reasons 5. Given that: UV XY, VW WX, WX YZ, prove UW XZ Complete the proof by correctly ordering the reasons to correspond with the correct statements. Statements 1. UV XY, VW WX, WX YZ 2. VW YZ 3. UV = XY, VW = YZ 4. UV + VW = XY + YZ 5. UV + VW = UW, XY + YZ = XZ 6. UW = XZ 7. UW XZ Reasons 1 2 3 4 5 6 7 Reason choices Transitive property of congruence Addition (property of equality) Definition of congruent segments Given Segment addition postulate Definition of congruent segments Substitution property 6. Given the diagram at right, prove that AC = 2 BC. Statements Reasons
Geometry-reteaching angle relationships Name W e2c0[1m6i wknuztka\ hsmoufztpwoabrzew `LYLHCR.H T XAVlPlz _rxi`gbhatbsz trvecskerrnvleld]. 2.8 Angle Relationships Name the relationship: complementary, linear pair, vertical, or adjacent. 1) 2) a b a b 3) 4) b a b a Find the measure of angle b. 5) 6) b 112 63 b 7) 8) 55 b 35 b ^ p2c0s1[6m dkqujtxah osjoqfztpwhadrze_ ylvlrcu.z N mafljli druivgqhmtjsn BrGeosieqr`vteZdx.\ R ]MdandUe^ ywwidtihf pitnyf^imneiqtje] ng`emoymleitmrxyx. -1- Worksheet by Kuta Software LLC
How are these angles related? Make an equation then find the value of x. 9) 10) (x + 17) 36 (5x + 1) (3x + 1) 11) 12) (3x + 2) 80 5x 38 13) 14) 299 66 25 (x - 13) (x + 2) 15) 16) 2x 40 (4x + 2) 82 H f2u0u1f6d dkauntzal KS[omfVt]wVaKrSet ZLxLqCR.g N fahltla Fr[iNgghMtBsV [rsessxeyrqvoehdp.w P ^Mna[d[es uwmi\tdhr EIDnWfciknwi`tDeV RGPeUoVm]e\tzrWyv. -2- Worksheet by Kuta Software LLC
Name: Period: 2.9 Quiz Unit 2 Review 1.) Explain the difference between inductive and deductive reasoning. You may use an example to help explain the difference between the two. 2.) Find counter-examples to the following statements to prove them false. a) All angles are acute b) 3 x is a prime number for all values of x. c) All rational numbers have a decimal portion. 3.) Complete the patterns by drawing the next 2 steps. a) b) c) 4) Complete each sequence by writing in the next 2 terms a) -1, 4, -9, 16, -25,, b) 1, 2, 3, 4,, 2 3 4 5 c) 1, 3, 6, 10,, d) 1, 5, 12, 22, 35,, 5) Complete the KenKen Puzzle
6) Prove that if 2 angles form a linear pair and are congruent, then they must be right angles. Hint: how many degrees go around a full rotation? 7) 8) 9) 10) 11) Complete the proof. (hint: how can you use the definition of supplementary?)