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Transcript
Lesson 1.1 Patterns and Inductive Reasoning You will learn to… * find and describe patterns * use inductive reasoning to make conjectures Sketch the next figure. 1. 14 2. 2 4 8 16 Describe the pattern. Find the next three numbers. 3. 0, 1, 1, 2, 3, 5, 8,… 13, 21, 34 4. 4, 9, 16, 25, … 36, 49, 64 5 , __ 6 , __ 7 5. __ 1 , __ 2 , __ 3 , __ 4 , __ 2 3 4 5 6 7 8 How many squares are in the next object? 6. 3, 6, 10, 15 7. 1, 4, 9, 16 A conjecture is an unproven statement that is based on a pattern. Inductive Reasoning is the process of looking for a pattern and making a conjecture. Complete the conjecture. 8. The sum of any 2 odd even numbers is __________. 9. The product of any 2 odd odd numbers is __________. A counterexample is an example that shows a conjecture is false. Find a counterexample. 10. The sum of 2 numbers is always greater than the larger of the numbers. 11. If a shape has 2 sides the same length, then it must be a rectangle. 12. If a shape has 4 sides, then it must be square. Lesson 1.2 Points, Lines, and Planes You will learn to… * understand and use the basic geometry terms * sketch intersections of lines and planes Undefined terms cannot be mathematically defined using other known words. point line plane Two points determine a line. Postulate 2 Three points determine a plane. B A T C plane ABC or plane T A B D H E C G F Do you see… plane ABF ? plane ADG? Collinear points A points that lie on the same line Coplanar points points that lie in the same plane Coplanar lines lines that lie in the same plane C B B A C Betweeness refers to collinear points only. Point B is between A and C. A B C Line AB C n A B CA BA AB line n BC Segment AB B A AB BA Is AC the same as AB? B A C NO Ray AB B A AB BA Opposite Rays share an endpoint A NA N B ? NB a line Opposite rays form __________. 1) Draw three noncolinear points J, K, and L. 2) Draw JK, KL, and LJ. K J L If 2 lines intersect, then their a point intersection is ____________. If 2 planes intersect, then their a line intersection is ___________. If a line intersects a plane they intersect at __________. a point Lesson 1.3 Segments & Their Measures You will learn to… * use segment postulates * use the distance formula The distance between points A and B is written as AB which is the length of AB. A -3 AB = B -2 -1 0 1 2 3 | 2 – 3| or |3 – - 2| Distance is the absolute value of their difference. =5 1. Find the length of the segment. 6 3 inches =1 1 8 4 Draw a segment that is…. 3. 4 cm long 4. 2.7 cm long 5. 56 mm long A postulate is a statement or rule that is accepted without proof. Rules that are proven are called theorems. Segment Addition Postulate If B is between A and C, then AB + BC = AC. AB A BC B AC C 6. Find AB. 20 A B 50 AB + 20 = 50 AB = 30 C 7. Write an expression for AC. 2x - 5 3x + 2 A B C ? AC = (2x – 5) + (3x + 2) AC = 5x - 3 8. Write an expression for AC. 8x + 1 A C E 12x + 10 AC + (8x + 1) = 12x + 10 AC = 4x + 9 9. Suppose M is between L and N. Use the Segment Addition Postulate to solve for x. LM = 3x + 8 L M N MN = 2x – 5 LN = 23 3x + 8 + 2x – 5 = 23 5x + 3 = 23 5x = 20 x=4 10. E is between H and R. A is between E and R. R is between E and T. HT = 50, ER = 20, and HE=EA=AR. Find RT. 50 10 10 10 H E A R 20 T EA = AR = 10 RT = 20 HE = 10 11. Find AB and CD. C A B D AB= 5 CD= 5 Congruent Segments have the same lengths. AB = CD AB CD lengths are equal segments are congruent 12. Use the Pythagorean Theorem to find the distance between the points. (4,6) 2 c= 4 (1,2) 3 2 (4-1) 2 (6-2) ? + 2 2 (x2-x1) + (y2-y1) 2 c= 2 3 + 2 4 9 + 16 c2= 25 c= 5 Distance Formula If A (x1, y1) and B (x2, y2) are points in a coordinate plane, then AB = (x2 – x1 + (y2 – y1 2 ) 2 ) 13. Find TM T (3, -2) and M (1, 4) TM = TM = 2 (1 - 3) 2 (-2) + + 2 (4 - 2) 2 (6) TM = 4 + 36 = 40 ≈6.32 4 · 10 = 2 10 Tiled Floor Street Corner Two-Point Perspective Street Corner Two-Point Perspective Lesson 1.4 Angles and Their Measures You will learn to… * use angle postulates * classify angles as acute, right, obtuse, or straight An angle consists of 2 rays that have the same endpoint called the vertex of the angle. X Y YX and YZ form XYZ 60° Z 1. Name the sides, vertex, and angle. Sides: BA and BC Name: ABC Vertex: B A CBA B B 54° C Congruent Angles have the same measure. mABC = mXYZ ABC XYZ equal measures congruent angles Classifying Angles By Their Measures… Acute Angle Measure is between 0˚ and 90˚ Classifying Angles By Their Measures… Obtuse Angle Measure is between 90˚ and 180˚ Classifying Angles By Their Measures… Right Angle Measure is 90˚ Classifying Angles By Their Measures… Straight Angle Measure is 180˚ Two angles are Adjacent Angles if they share a vertex and a side but have no common interior points. (not overlapping) C A B ABC and CBD are adjacent angles D 2. List the 3 angles shown. Which angles are adjacent? R P S T RSP RSP PST PST RST Angle Addition Postulate R P S T m RSP m PST m RST 3. Find m RST. R S 35º 94° P 59º T 4. Find m RSP. 60° 110º R P S 50º T How do you measure an angle with a protractor? 5. Use a protractor to draw a 65˚ angle. 6. Use a protractor to draw a 112˚ angle. 45° or 55°? 145° or 135°? 45° or 55°? 145° or 135°? 35° or 25°? 155° or 165°? 35° or 25°? 155° or 165°? 120° - 90° mNPR = 30° 90° - 60° HW: 29-34 C B A D E F Artist: Julian Beever People are actually avoiding walking in the "hole" 'Make Poverty History' drawing from the side (40 ft long) Lesson 1.5 Segment and Angle Bisectors You will learn to… * bisect a segment * bisect an angle A midpoint is a point that divides a segment into two congruent segments. M A midpoint B AM = MB AM MB To bisect a segment means to divide it into two congruent segments. Use a compass to locate the midpoint of a segment. A segment bisector intersects a segment at its midpoint. How would you find the “middle” between 2 numbers? How would you find the “middle” between the points? Midpoint Formula If A (x1, y1) and B (x2, y2) are points in a coordinate plane, xx , y y 2 2 1 2 1 2 1. Find the midpoint between (-2, 3) and (4, -6) -2 + 4 , 3 + -6 2 2 (-2,3) (1, -1.5) (4,-6) 2. Find the midpoint between (2, -1) and (4, -4) 2 + 4 , -1 + -4 2 2 (3, -2.5) 3. One endpoint is (-3, -1). The midpoint is (3, -4). Find the other endpoint. (-3,-1), (3,-4), (x,y) (9, -7) -3 + x 3 = 31 2 -1 + y -4 = -4 1 2 -3 + x = 6 x=9 -1 + y = -8 y = -7 3. One endpoint is (3, -5). The midpoint is (-2, 4). Find the other endpoint. (3,-5), (-2,4), (x,y) (-7,13) 3+x = -2 2 3 + x = -4 x = -7 -5 + y 4 = 2 -5 + y = 8 y = 13 An angle bisector is a ray that divides an angle into 2 congruent angles. angle bisector 4. NQ is an angle bisector. Find m MNQ. mMNQ = 57° Q M 114° N P 5. NQ is an angle bisector. Find m MNQ and m MNP. mMNQ = 64° Q M mMNP = 128° 64° N P 6. NQ is an angle bisector. Find x. 2x – 5 = 55 Q M x = 30 55° (2x – 5)° N P 7. BD bisects ABC. m ABD = (2x + 50)˚ m DBC = (5x + 5)˚ Find the measure of all 3 angles. 5x+5 = 2x+50 3x = 45 x = 15 mABC = 160° A D 80° 80° B C Workbook Page 13 (1-8) Lesson 1.6 2 Angle Pair Relationships You will learn to… * identify vertical angles and linear pairs * identify complementary and supplementary angles Two angles are Complementary Angles if the sum of their measures is 90 A 90˚ angle is a right angle and forms a corner. Two angles are Supplementary Angles if the sum of their measures is 180 A 180˚ angle forms a straight angle. m A = 25° 1. Find its complement. 65° 2. Find its supplement. 155° Do complementary angles have to be adjacent? 50° 10° 40° 80° adjacent nonadjacent Do supplementary angles have to be adjacent? 70° 105° 110° 75° adjacent nonadjacent 3. Are the two angles formed complementary or supplementary? 120° 60° supplementary angles Two angles are vertical angles if they are NOT adjacent and their sides are formed by 2 intersecting lines. vertical angles 1 3 4 2 1 and 2 3 and 4 Two angles form a linear pair if they are adjacent angles whose noncommon sides form a line. 1 3 4 2 linear pairs 1 and 4 1 and 3 2 and 4 2 and 3 Which angles are… supplementary? congruent? 1 and 4 1 and 3 2 and 4 2 and 3 1 and 2 3 and 4 1 4 3 2 Linear Pair Postulate If two angles form a linear pair, then they are _____________. supplementary Vertical Angles Theorem Vertical angles are congruent ___________. 4. Find the measure of each angle. m1 = 140° 1 3 40° 2 m2 = 140° m3 = 40° 5. Find x and y. Then find the angle measures. 5x+30+4x+15=180 105° 3y-15+3y+15=180 75° (5x + 30)° (3y - 15)° (4x + 15)° (3y + 15)° 75° x = 15 105° y = 30 Lesson 1.7 Perimeter & Area You will learn to… * find the perimeter and area of common plane figures Area How many squares will fit inside a region? Perimeter What is the distance around a region? Square Area = A= 2 side 2 s Perimeter = 4(side) P = 4s Rectangle Area = base ( height ) A = bh Perimeter = 2(base) + 2(height) P = 2b + 2h 1. Find the area of a square that has a side length of 20 inches. Find the perimeter. 2 20 A= 2 A = 400 inches P = 4(20) P = 80 inches 2. Find the area of a rectangle that is 20 m by 4 m. Find the perimeter. A = 20(4) 2 A = 80 m P = 2(20) + 2(4) P = 48 m 3. A rectangle has an area of 98 cm2. Find the length of its base if its height is 7 cm. A = b(h) 98 = b(7) b = 14 cm Triangle Area = ½ base ( height) A=½ b h Perimeter = sum of 3 sides P= a + b + c Find the area and perimeter of the triangle. P = 40.8 cm 4. 9.8 cm c2 = 4 2 + 9 2 4 9 cm 15 cm 152 = x2 + 92 12 16 cm 9(16) 2 A= = 72 cm 2 5. Find the area and perimeter of the triangle. A= 9(6) 2 9 cm ?9.8 13.5 ? A = 27 cm2 6 cm 10 cm P = 29.3 cm 2 c = 2 9 + 2 2 410 Find the area of the triangle. 6. 25 cm 25(12) A= 2 12 cm 20 cm A = 150 2 cm Find the area of the triangle. Pythagorean Theorem? 6. 15 cm 25 cm 20(15) A= 2 12 cm 20 cm A = 150 2 cm center radius diameter diameter = 2 (radius) Circumference of a Circle C = (diameter) π C = (2 · radius) π C= dπ or C=2rπ 7. Find the circumference. 6 cm C = 12π cm ≈ 37.7cm 8. Find the circumference. C = 10π cm ≈ 31.4 cm 10 cm 9. Find the radius of the circle that has a circumference of 120 feet. C = dπC = 2rπC = 2rπ 120 = 2rπ 2π 2π 19.1≈ r Area of a Circle 2 A = (radius) π A= 2 r π 10. Find the area. 6 cm A = 36π cm2 A ≈ 113.1 cm2 11. Find the area. A = 64π 2 cm A ≈ 201.1 2 cm 16 cm 12. Find the radius of the circle that has an area of 120 square feet. 2 r π 120 = π π 2 r = 38.197 r = 38.197 r ≈ 6.2 ft 13. Find the perimeter and area of the triangle. -1) 22 AB = (3 BC (-2---2) 3)22++(-1 (-4--2) AC = 6 AB = 34 ≈ 5.8 BC = 34 ≈ 5.8 height = 5 (-2,2) C P = 6 + 2 34 ≈ 17.6 A = ½ (5)(6) = 15 square units B (3,-1) A (-2,-4) 14. Find the area of the bluish region. Square? 2 A = 12 = 144 12 cm Circle? A = 62 π = 36π Square-Circle = 144 - 36π ≈ 30.9 cm2 15. Find the area of the bluish region. r=5 Square? 2 A = 20 = 400 Circle? A = 52 π = 25π Square-Circles = 400 – 4(25π) ≈ 85.84