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Section Li Patterns and Inductive Reasoning Inductive Reasoning: Conjecture: Example 1: Finding and Using a Pattern Find a pattern for each sequence. Use the pattern to show the next two terms of the sequence. a. 3,6,12,24,... b. • N 7 \j_i / /\ i \__v Example 2: Using Inductive Reasoning Make a conjecture about the sum of the first 30 odd numbers. Counterexample: Example 3: Finding a Counterexample Find a counterexample for each conjecture. a. The square of any number is greater than the original number. b. You can connect any three points to form a triangle. c. Any number and its absolute value are opposites. Example 4: Real-World Connection A skateboard shop finds that over a period of five consecutive months, sales of smallwheeled skateboards decreased. Use inductive reasoning to make a conjecture about the number of small-wheeled skateboards the shop will sell in June. Skateboards Sold - 601 55j 50j 45 1 J r M A M Month 2 _____ ______ ______ Nets for Three-Dimensional Figures Section 1.2 Net: Example 1: Identifying Solids From Nets Given the net below, which figure will be formed by folding the net? a. c b. d Example 2: Drawing a Net Draw a net for the graham cracker box. Label the net with its dimensions. :4 Section 1.3 Points, Lines, and Planes Point: Line:. B -—4-- Collinear points: Example 1: Identifying Collinear Points a. Are points E, F, and C collinear? If so, name the line on which they lie. b. Are points E, F, and D collinear? If so, name the line on which they lie. Plane: p Coplanar: B A Example 2: Naming a Plane Each surface of the ice cube represents part of a plane. Name the plane represented by the front of the ice cube. H F D C- A 4 Postulate: Postulate 1-2: If two lines intersect, then they intersect in exactly one point. p Example 3: Finding the Intersection of Two Planes What is the intersection of plane HGFE and plane BCGF? A B Postulate 1-4: Through any thiee noncollinear points there is exactly one plane. Example 4: Using Postulate 1-4 Name the plane that is represented by the shaded area. a. b. Ni \ Section 1.4 Segments, Rays, Parallel Lines and Planes Segment: - -a--——- —-—. - - Ray: Opposite Rays: p _.—__-__:_ _- R Example 1: Naming Segments and Rays Name the segments and rays in the figure. Segments: Rays: - Parallel lines: Skew lines Parallel planes: Example 2: Identifying Parallel and Skew Segments and Parallel Planes a. Name all segments that are parallel to DC: b. Name all segments that are skew to DC: H B c. Name two pairs of parallel planes: d. Name a line that is parallel to plane GHIJ: 6 _____________ Section 1.5 - Measuring Segments Ruler Postulate The points of a line can be put into one-to-one correspondence with the real numbers so that the distance between any two points is the absolute value of the difference of the corresponding numbers. B L - Congruent() ( Congruent segments: i’ Xb i1; IQ •k--: C Example 1: Comparing Segment Lengths — Find AR and BC. Are AR and BC congruent? —A I} •1- I 91 fl —3----(——4——2—I 0 1 2 ‘ - I Segment Addition Postulate: If three points A, B, and C are collinear and B is between A and C, then B .. Example 2: Using the Segment Addition Postulate If DT = 60, find the value ofx. Then find DS and ST. N c T 7 _______-- Midpoint: Example 3: Using the Midpoint C is the midpoint of AB. Find AC, CB, and AR 2 + I p B 8 Section 1.6 Measuring Angles Angle (L): I r . Example 1: Naming Angles Name the following angle four different ways. C _— A Classifying angles: Example 2: Measuring and Classifying Angles Using a protractor, find the measure of each angie. Classify as either acute, obtuse, right, or straight. a. I // b. C. - 9 Congruent angles: Angle Addition Postulate: If point B is in the interior ofL4OC then . •4\ “B , f-i ( If L4OC is a straight angle, then — A Example 3: Using the Angle Addition Postulate a. What is LTSW if mZRST = 50 and mLPSW = 125? b. Suppose that m1 =42 and mL4BC =88. Find mL2. 1 A // B • c 10 ... 0 ( Angle pairs: Z4 - 4 ii;: 4 / / / 4 4 Example 4: Identifying Angle Pairs In the diagram identify pairs of numbered angles that are related as follows: a. Complimentary I b. Supplementary c. Vertical Example 5: Making Conclusions From a Diagram What can you conclude from the information in the diagram? / / 3/ —— 11 Section 1.7 - Basic Constructions Example 1: Constructing Congruent Segments Construct a segment congruent to the given segment: Il Step 1: Using a straightedge, draw a ray with endpoint C Step 2: Open the compass to the length of AB A B Step 3: With the same compass setting, put the compass point on point C. Draw an arc that intersects the ray. Label the point of intersection D. 12 Example 2: Constructing Congruent Angles Construct an angle congruent to the given angle: Step 1: Draw a ray with endpoint S. Step 2: With the compass on point A, draw an arc that intersects the sides ofLA. Label the points of intersection B and C. A ‘- Step 3: With the same compass setting, put the compass point on point S. Draw an arc and label its point of intersection with the ray as R. S Step 4: Open the compass to the length of BC. Keeping the same compass setting, put the compass point on 1?. Draw an arc to locate point T. c Step 5: Draw ST. I R 13 Perpendicular lines: c\ /\\\ Perpendicular bisector: B Example 3: Cdnstructing the Per,endicular liseêtor Construct the perpendicular bisector of segment AB: - U. Step 1: Put the compass on point A and draw a long arc as shown. Be sure the opening is greater than half of — A — B Step 2: With the same compass setting, put the compass point on point B and draw another long arc. Label the points where the two arcs intersect as X and Y. A B •1 Step 3: Draw XY. The point of intersection of AB and XY is M, the midpoint of AR. 4 . II V Li Angle bisector: Example 4: Finding Angle Measures KíV bisects LJKL so that mLJKJ’/ = 5x —25 and mLNKL mLJKN. / ‘, = 3x +5. Solve for x and find r , Example 5: Constructing the Angle Bisector Construct the bisector of L4: Step 1: Put the compass point on vertex A. Draw an arc that intersects the sides of LA. Label the points of intersection BandC Step 2: Put the compass on point C and draw an arc. With the same compass setting, draw an arc using point B. Be sure the arcs intersect. Label the point where the two arcs intersect asX Step 3: Draw AX. V. Section 1.8 The Coordinate Plane f_ I I —) (÷ The Distance Formula: The distance d between two points A(x 1 y) and 2 B(x 2 y is ) , , I Example 1: Finding Distance Find the distance between T(5, 2) and R(-4, -1) to the nearest tenth. Example 2: Each morning Juanita takes the “Blue Line” subway from Oak Station to Jackson Station . As the maps below shows, Oak Station is 1 mile west and 2 miles south of City Plaza. Jackson Station is 2 miles east and 4 miles north of City Plaza. Find the distance Juanita travels between Oak Station and Jackson Station. I-’ 4 —4 —.- r llt.t (U The Midpoint Formula: The coordinates of the midpoint M of AB with endpoints A(x 1 following: — , ) 1 y and 2 B(x y,) are the , Example 3: Finding the Midpoint QS has endpoints Q(3, 5) and S(7, -9). Find the coordinates of its midpoint M Example 4: Finding an Endpoint The midpoint of AB is M(3, 4). One endpoint is A(-3, -2). Find the coordinates of the other endpoint B. Section 1.9 Perimeter Circumference and Area Perimeter and Area: / . ,/.‘ I) I .1 N• Square with side length s Rectangle with base b and height h Circle with radius r and diameter d Perimeter P = Perimeter P Circumference C AreaA AreaA = = AreaA = =‘- fi Example 1: Your pool is 15 ft wide and 20 ft long with a 3-ft wide deck surrounding it. You want to build a fence around the deck. How much fencing will you need? - \ ___4 -‘= —— - — - b / . : -- i Example 2: Finding Circumference Find the circumference o( A in terms ofg. Then find the circumference to the nearest 5 tenth. / LA >4 \. \/*. A1: 12 1 - Example 3: Finding Perimeter in the Coordinate Plane Find the perimeter of AABC. ii tiii r2 c 11 V -. 1— I! / / — A C ‘LA 2 .A: x —, no.) jy c_ti. 4/ I IF I — — 4, 1 A- iaYiE — ::•i/’4 F A I [t V:c1: — A I —t —I tThC t(. 1 ‘5: I, <; A . ‘I •4, ‘4’ . Cr .:4,E1 I —; C -. — — A ‘1 1. IA! o :I.Il. ... A 4 A Example 4: Finding Area of a Rectangle You are designing a rectangular banner for the front of a museum. The banner will be 4 ft wide and 7 yd high. How much material do you need? Example 5: Finding Area of a Circle The diameter of a circle is 10 in. Find the area in terms of Example 6: Finding Area of an Irregular Shape What is the area of the figure below? ir.