text stringlengths 235 3.08k |
|---|
D is a point on side and E is a point on side. If AD 6, DB 9, and AC 20, find AE and EC. AB AC DE BC of ABC such that 12. A pile of gravel is in the form of a cone. The circumference of the pile of gravel is 75 feet and its height is 12 feet. How many cubic feet of gravel does the pile contain? Give your answer to the... |
ent, prove that the quadrilateral is a trapezoid. intersect at E. If ABE and BD c. Triangle ABC is equilateral. From D, the midpoint of AB, and to F, the midpoint of, a line BC segment is drawn to E, the midpoint of AC. Prove that DECF is a rhombus. 14365Index.pgs 7/13/07 10:19 AM Page 636 INDEX A AA triangle similarit... |
, 16 naming of, 16 obtuse, 17 plane, 424 postulates of, 135–138 proving theorems about, 144–145 right, 17 sides of, 15 straight, 16, 17 subtraction of, 20–21 sum of measures of, of triangle, 347–350 supplementary, 145–146, 148–149 theorems involving pairs of, 146–148 trisection of, 604 using congruent triangles to prov... |
379 Boole, George, 34 Boolean algebra, 34 C Cartesian coordinates, 209 Cavalieri, Bonaventura, 419 Cavalieri’s Principle, 419 Center of circle, 460, 536 chords equidistant from, 546–550 of regular polygon, 450 of sphere, 459 Center-radius equation of a circle, 582 Centimeter, cubic, 446 Central angle of circle, 536–53... |
–57, 60, 335 contrapositive of, 64–65 converse of, 63–64 false, 63, 65 hidden, 55–57, 98 inverse of, 61–62 parts of, 55 as they relate to proof, 138–139 true, 63, 64 truth value of, 54–55 Cone(s), 456–458 altitude of, 456 base of, 456 frustum of, 459 height of, 456 right circular, 456 slant height of, 456 surface area ... |
591 tangents in, 588 translations in, 232–235 preservation of angle measure, 232 preservation of collinearity, 232 preservation of distance, 232 preservation of midpoint, 232 Coordinate proof, 313–314 of general theorems, 313 for special cases, 313 Coordinates, 210 of point in plane, 210–213 rectangular, 211 Coplanar l... |
124, 270–271 Dk, 247, 496 Domain, 36, 250 E Edge of polyhedron, 440 Elements (Euclid), 1, 262, 474 Endpoint, of ray, 15 Enlargement, 496 Epicureans, 262 Equality reflexive property of, 110–111 symmetric property of, 111 transitive property of, 111, 263–264 Equation(s) of line, 295–299 solving, with biconditionals, 70–... |
, 517–518 Frustum of cone, 459 Function(s) defined, 250 transformations as, 250–254 G Galileo, 419 Generalization, 94 General quadrilateral, 380 Geometric constructions, 196. See also Constuctions Geometric inequalities, 262–285 basic inequality postulates, 263–265 inequalities involving lengths of the sides of a trian... |
5 involving exterior angle of triangle, 276–279 involving lengths of sides of triangle, 273–274 Lateral sides of prism, 440 Lateral surface of cylinder, 453 Law(s) DeMorgan’s, 34 of Detachment, 75, 101, 105 of Disjunctive Inference, 76 of logic, 35, 74–78 Leg(s) proving right triangles congruent by hypotenuse, 362–365 ... |
quadrilateral, 379 Isosceles trapezoid(s) base angles of, 403, 404 properties of, 403 proving that quadrilateral is, 403–407 Isosceles triangle(s), 24, 25, 181–183, 451 base angle of, 25 parts of, 25 vertex angle of, 25 Isosceles triangle theorem, 181 converse of, 357–360 L Lateral area of prism, 442 Lateral edges of ... |
67 Lower base angles of isosceles trapezoid, 404 M Major arc, 537 Mathematical sentences, 35 Mean(s), 476 geometric, 478 Mean proportional, 478–479, 512 Measure of angle, 16, 17, 552–555 formed by tangent intersecting a secant, 568 formed by two intersecting chords, 568 formed by two intersecting secants, 569 formed by... |
Origin, 210 Orthocenter, 319 P Parabola, turning point of, 626 Paragraph proof, 101–102 Parallelepiped(s), 441, 442 rectangular, 442 volume of, 446 Parallel lines, 328–372, 421 constructing, 605–606 in coordinate plane, 342–344 defined, 329 equidistant from two, 621 methods of proving, 333 planes and, 433–439 properti... |
410 circumscribed about a circle, 563–564 concave, 368 convex, 368 diagonals of, 368 exterior angles of, 276–277, 369–371 graphing, 212–213 inscribed in circle, 550–551 interior angles of, 368–369 regular, 369 sides of, 23 similar, 486–488 Polyhedron(s), 440–442 bases of, 440 edges of, 440 faces of, 440 vertices of, 44... |
SSS, 489–494 SAS, 489–494 Pyramid(s), 449 altitude of, 449 height of, 449 properties of regular, 450–451 regular, 449 slant height of, 449 surface area of, 449 vertex of, 449 volume of, 449 Pythagoras, 1, 474 Pythagorean Theorem, 474, 515 converse of, 516–517 Pythagorean triple, 517 Q Quadrilateral(s), 23, 212, 379–41... |
-45 degree, 517–518 proportions in, 510–513 proving congruent by hypotenuse, leg, 362–365 Pythagorean theorem and, 515–517 30-60 degree, 518 rk, 217 RO, 230 Roots postulate, 125–126 Rotation in coordinate plane, 238–239, 240–242 defined, 238 negative, 239 positive, 239 preservation of distance under, about a fixed poin... |
ant height of cone, 456 of pyramid, 449 Slope of line, 291–294 negative, 293 of perpendicular lines, 307–310 positive, 292 undefined, 293 zero, 293 Slope-intercept form of an equation, 298 Solid geometry, 420 Solid, rectangular, 442–443 Solution set, 36 Sphere, 459–463 center of, 459 great circle of, 460 radius of, 459... |
ence, 362–365 involving pairs of angles, 146–148 Isosceles Triangle, 181 converse of, 357–360 Midsegment, 480 Perpendicular Bisector Concurrence, 193 proving, about angles, 144–145 Pythagorean, 474, 515 converse of, 515–517 Triangle Inequality, 273 Triangle Similarity, 490 30-60-degree right triangle, 518 Total surface... |
HL, 362–365 proving similar by AA, 489–494 SSS, 489–494 SAS, 489–494 right, 25, 26, 515 45-45 degree, 517–518 proportions in, 510–513 Pythagorean theorem and, 515–517 30-60 degree, 518 scalene, 24, 25, 98 sum of measures of angles of, 347–350 Triangle Inequality Theorem, 273 Triangle Similarity Theorem, 490 Triangular... |
is 18. Complete the 5-by-5 magic square on the right. Use only the numbers in this list: 6, 7, 9, 13, 17, 21, 23, 24, 27, and 28. 5 4 9 10 6 2 3 8 7 20 29 22 19 12 10 25 18 11 8 14 26 30 15 16 12 CHAPTER 0 Geometric Art L E S S O N 0.4 Everything is an illusion, including this notion. STANISLAW J. LEC Op Art Op art, o... |
a compass and straightedge (and doing some careful coloring). Can you figure out how each of these op art designs was created? 14 CHAPTER 0 Geometric Art EXERCISES 1. What is the optical effect in each piece of art in this lesson? 2. Nature creates its own optical art. At first the black and white stripes of a zebra a... |
collection of Celtic knot designs. L E S S O N 0.5 In the old days, a love-sick sailor might send his sweetheart a length of fishline loosely tied in a love knot. If the knot was returned pulled tight it meant the passion was strong. But if the knot was returned untied— ah, matey, time to ship out. OLD SAILOR’S TALE C... |
points in the sand with their fingertips, as shown below left. Then they begin to tell a story and, at the same time, trace a finger through the sand to create a lusona design with one smooth, continuous motion. Try your hand at creating sona (plural of lusona). Begin with the correct number of dots. Then, in one moti... |
own sketch or painting to evoke a feeling or to tell a story. Write a one- or two-page story related to your art. ISEKI/PY XVIII (1978), Kunito Nagaoka LESSON 0.5 Knot Designs 19 Islamic Tile Designs Islamic art is rich in geometric forms. Early Islamic, or Muslim, artists became familiar with geometry through the wor... |
down into the lines of their design. When the tiling is complete, artists pour concrete over the tiles to form a slab. When the concrete dries, they lift the whole mosaic, displaying the colors and connected shapes, and mount it against a fountain, palace, or other building. EXERCISES 1. Name two countries where you c... |
one of the designs you made in Exercises 6 and 7. Trace or photocopy several copies and paste them together in a tile pattern. (You can also create your tessellation using geometry software and print out a copy.) Add finishing touches to your tessellation by adding, erasing, or whiting out lines as desired. If you wan... |
three things in nature that have geometric shapes. Name their shapes. 7. Draw an original knot design. 8. Which of the wheels below have reflectional symmetry? Hot Blocks (1966–67), Edna Andrade How many lines of symmetry does each have? Wheel A Wheel B Wheel C Wheel D 9. Which of the wheels in Exercise 8 have only ro... |
Does the flag of Kenya have rotational symmetry? Explain. c. Name a country whose flag has both rotational and reflectional symmetry. Sketch the flag. Assessing What You’ve Learned KEEPING A PORTFOLIO This section suggests how you might review, organize, and communicate to others what you’ve learned. Whether you follo... |
B. V.–Baarn–Holland. All rights reserved In this chapter you will ● write your own definitions of many geometry terms and geometric figures ● start a notebook with a list of all the terms and their definitions ● develop very useful visual thinking skills L E S S O N 1.1 Nature’s Great Book is written in mathematical s... |
ancient China said, “The line is divided into parts, and that part which has no remaining part is a point.” Those definitions don’t help much, do they? A definition is a statement that clarifies or explains the meaning of a word or a phrase. However, it is impossible to define point, line, and plane without using word... |
, between equal numbers and the congruence symbol,, between congruent figures. C 3.2 cm 3.2 cm A D AC DC AC DC When drawing figures, you show congruent segments by making identical markings. These single marks mean these two segments are congruent to each other. B A C S P These double marks SP RQ, mean that and these t... |
7, draw two points and label them. Then use a ruler to draw each line. Don’t forget to use arrowheads to show that it extends indefinitely. 5. AB 6. KL 7. DE with D(3, 0) and E(0, 3) For Exercises 8–10, name each line segment. A 8. C 9. y 5 P 10. R I Q x 10 T For Exercises 11 and 12, draw and label each line segment. 1... |
(2, 2). 31. Draw CD through points C(2, 1) and D(2, 3). y 5 –5 x A (4, 0) –5 Career Woodworkers use a tool called a plane to shave a rough wooden surface to create a perfectly smooth planar surface. The smooth board can then be made into a tabletop, a door, or a cabinet. Woodworking is a very precise process. Producing... |
points of a segment, x1 x2, y1 2 2 y2 History Surveyors and mapmakers of ancient Egypt, China, Greece, and Rome used various coordinate systems to locate points. Egyptians made extensive use of square grids and used the first known rectangular coordinates at Saqqara around 2650 B.C.E. By the seventeenth century, the ag... |
the midpoint of BD in each figure. b. What do you notice about the midpoints? y A B B 10 5 B C (35.5, 10) C (22, 8.5) Figure 1 Figure 2 Figure 3 A D 5 C 10 D (16, 0.5) A 15 20 25 D (29.5, 1) 30 35 x USING YOUR ALGEBRA SKILLS 1 Midpoint 37 L E S S O N 1.2 Inspiration is needed in geometry, just as much as in poetry. AL... |
big to you? Which seem small? The measure of an angle is the smallest amount of rotation about the vertex from one ray to the other, measured in degrees. According to this definition, the measure of an angle can be any value between 0° and 180°. The smallest amount of rotation PG to PS from is 136°. 136° P 224° G S Th... |
. By how many degrees does the Sun’s position in the sky change every hour? 40 CHAPTER 1 Introducing Geometry EXAMPLE C Look for angle bisectors and congruent angles in the figures below. a. Name each angle bisector and the angle it bisects. b. Name all the congruent angles in the figure. Use the congruence symbol and ... |
Introducing Geometry B C D E F 6. Draw a figure that contains at least three angles and requires three letters to name each angle. For Exercises 7–14, find the measure of each angle. B C Y 0 30 40 5 0 6 150 140 90 100 110 12 80 70 1 0 0 90 0 1 3 0 60. mAQB? 8. mAQC? 9. mXQA? 10. mAQY? 11. mZQY? 12. mZQX? 13. mCQB? 14.... |
use your protractor or your ruler. 34. MI? IC? mM? Y E K M C I 35. MEO? SUE? OU? S U E M O 44 CHAPTER 1 Introducing Geometry For Exercises 36–38, do not use a protractor. Recall from Chapter 0 that a complete rotation around a point is 360°. Find the angle measures represented by each letter. 36. 37. 38. 15° 21° x 1 4... |
solution by listing in order which coin is moved. For example, your list might begin PDP.... 46 CHAPTER 1 Introducing Geometry L E S S O N 1.3 “When I use a word,” Humpty replied in a scornful tone,“it means just what I choose it to mean—neither more nor less.” “The question is,” said Alice,“whether you can make a wor... |
can probably find more than one!) b. Write a better definition for a square. LESSON 1.3 What’s a Widget? 47 Solution You probably noticed that “figure” is not specific enough to classify a square, and that “four equal sides” does not specify how it differs from the first counterexample shown below. a. Three counterexa... |
xample to one of your definitions, write a better definition. As a group, decide on the best definition for each term. As a class, agree on common definitions. Add these to your notebook. Draw and label a picture to illustrate each definition. Right Angle 90° 46° 105° Right angles Not right angles Acute Angle 89° 33° 9... |
. Use the appropriate marks to indicate right angles, parallel lines, congruent segments, and congruent angles. Use a protractor and a ruler when you need to. 1. Acute angle DOG with a measure of 45° 3. Obtuse angle BIG with angle bisector IE 5. PE AR 7. Complementary angles A and B with mA 40° 2. Right angle RTE 4. DG... |
QR TS. 24. A partial mirror reflects some light and lets the rest of the light pass through. In the figure at right, half the light from point A passes through the partial mirror to point B. Copy the figure, then draw the outgoing angle for the light reflected from the mirror. What do you notice about the ray of refle... |
squares are joined together side to side. (A complete side must touch a complete side.) Some of the smaller polyominoes are shown below. There is only one monomino and only one domino, but there are two trominoes, as shown. There are five tetrominoes— one is shown. Sketch the other four. Monomino Domino Trominoes Tetr... |
polygons are congruent polygons if and only if they are exactly the same size and shape. “If and only if ” means that the statements work both ways. If polygons are congruent, then corresponding sides and angles are congruent. If corresponding sides and angles are congruent, then polygons are congruent. For example, i... |
polygons can you find in this quilt? 56 CHAPTER 1 Introducing Geometry 17. Name the diagonals of pentagon ABCDE. For Exercises 18 and 19, use the information given to name the triangle that is congruent to the first one. D 18. EAR? A I 19. OLD? E R T N 20. In the figure at right, THINK POWER. a. Find the missing measu... |
rectangles? Review 35. Name a pair of complementary angles and a pair of vertical angles in the figure at right. 36. Draw AB, CD, and EF with AB CD and CD EF. 37. Draw a counterexample to show that this statement is false: “If a rectangle has perimeter 50 meters, then a pair of adjacent sides measures 10 meters and 15... |
angles and Special Quadrilaterals 59 Solution By studying the markings, you can tell that AB CD, JK JM, and STU XYZ. In this lesson you will write definitions that classify different kinds of triangles and special quadrilaterals, based on relationships among their sides and angles. Investigation Triangles and Special Q... |
also for its overall attractiveness? At the Acoma Pueblo Dwellings in New Mexico, how do sunlight and shadows enhance the existing shape formations? How many shapes make up the overall triangular shapes of these pyramids at the Louvre in Paris? 62 CHAPTER 1 Introducing Geometry Kite Kites Not kites Recreation Today’s ... |
angle measuring 40° 19. Draw a hexagon with exactly two outside diagonals. 20. Draw a regular quadrilateral. What is another name for this shape? 21. Find the other two vertices of a square with one vertex (0, 0) and another vertex (4, 2). Can you find another answer? Austrian architect and artist Friedensreich Hunder... |
some optical illusions with op art in Chapter 0. Some optical illusions are tricks—they at first appear to be drawings of real objects, but actually they are impossible to make, except on paper. Reproduce the two impossible objects by drawing them on full sheets of paper. Can you create an impossible object of your ow... |
to make clear which arc you mean—the first and last letters are the endpoints and the middle letter is any other point on the arc. P Minor arc AP Semicircle APD A O D Major arc PAD Arcs have a degree measure, just as angles do. A full circle has an arc measure of 360°, a semicircle has an arc measure of 180°, and so o... |
7. Name two tangents. 8. Name a point of tangency. 9. Name two types of vehicles that use wheels, two household appliances that use wheels, and two uses of the wheel in the world of entertainment. 10. In the figure at right, what is mPQ?? mPRQ 11. Use your compass and protractor to make an arc with measure 65°. Now ma... |
2x, 2y) 20. (x, y) → (2x, y5 x 5 –5 5 –5 –5 Review For Exercises 21–24, draw each kind of triangle or write “not possible” and explain why. Use your geometry tools to make your drawings as accurate as possible. 21. Isosceles right triangle 23. Scalene obtuse triangle 22. Scalene isosceles triangle 24. Isosceles obtuse ... |
well, you get the picture. Visualization skills are extremely important in geometry. So far, you have visualized geometric situations in every lesson. To visualize a plane, you pictured a flat surface extending infinitely. In another lesson, you visualized the number of different ways that four lines can intersect. Can... |
(1776–1831), a French mathematician with no formal education, wrote a prize treatise, contributed to many theories, and worked extensively on Fermat’s Last Theorem. The French mathematician Pierre de Fermat (1601–1665) developed analytic geometry. His algebraic approach is what made his influence on geometry so strong... |
Treasure here 40 m D Or here Initial diagram Improved diagram Final diagram LESSON 1.7 A Picture Is Worth a Thousand Words 75 EXERCISES 1. Surgeons, engineers, carpenters, plumbers, electricians, and furniture movers all rely on trained experience with visual thinking. Describe how one of these tradespeople or someone... |
locus of points in space that are equally distant from points A and B. 8. Draw an angle. Label it A. Sketch the locus of points in the plane of angle A that are the same distance from the two sides of angle A. 9. Line AB lies in plane. Sketch the locus of points in plane that are 3 cm from AB. Sketch the locus of poin... |
see the relationships among them. At right is a concept map showing the relationships among members of the triangle family. This type of concept map is known as a tree diagram because the relationships are shown as branches of a tree. Copy and fill in the missing branches of the tree diagram for triangles. 17. At righ... |
Fisk University Galleries, Nashville, Tennessee. 33. Use your compass to draw two congruent circles intersecting in exactly one point. How does the distance between the two centers compare with the radius? 34. Use your compass to construct two congruent circles so that each circle passes through the center of the othe... |
a two-dimensional drawing of a rectangular prism. This type of drawing is called an isometric drawing. It shows three sides of an object in one view (an edge view). This method works best with isometric dot grid paper. After practicing, you will be able to draw the box without the aid of the dot grid. Step 1 Step 2 St... |
7. The photo at right shows a prism-shaped building with a pyramid roof and a cylindrical porch. Draw a cylindrical building with a cone roof and a prismshaped porch. For Exercises 8 and 9, make a drawing to scale of each figure. Use isometric dot grid paper. Label each figure. (For example, in Exercise 8, draw the so... |
can pass through three noncollinear points. 23. If a line intersects a plane that does not contain the line, then the intersection is exactly one point. 24. If two lines are perpendicular to the same line, then they are parallel. 25. If two different planes intersect, then their intersection is a line. Physical models... |
’s the theory. Activity Chances Are In this activity you’ll see that you can apply probability theory to geometric figures. The Spinner After you’ve finished your homework and have eaten dinner, you play a game of chance using the spinner at right. Where the spinner lands determines how you’ll spend the evening. Sector... |
on them. How is geometric probability like the probability you’ve studied before? How is it different? Step 10 Create your own geometric probability problem. EXPLORATION Geometric Probability 1 87 ● CHAPTER 11 REVIEW ● CHAPTER 1 REVIEW ● CHAPTER 1 REVIEW ● CHAPTER 1 REVIEW ● CHAPTER CHAPTER 1 R E V I E W It may seem t... |
A knowledge of parallel lines, planes, arcs, circles, and symmetry is necessary to build durable guitars that sound pleasing. ● CHAPTER 1 REVIEW ● CHAPTER 1 REVIEW ● CHAPTER 1 REVIEW ● CHAPTER 1 REVIEW ● CHAPTER 1 14. A square is a rectangle with all the sides equal in length. 15. A pentagon has five sides and six dia... |
is the measure of A? Use your protractor. For Exercises 39–42, find the lengths x and y. y x 39. 42. 40. 1 3 8 3 2 4 4 18 y 7 x 2 2 2 20 A 41. y x x x 10 12 2y y 8 x y 12 43. If D is the midpoint of AC, C is the midpoint of AB, and BD 12 cm, what is the length of AB? 44. If BD is the angle bisector of ABC and BE is th... |
a full minute between jumps, how long will it take Jiminey to get home? 52. If the right triangle BAR were rotated 90° clockwise about point B, to what location would point A be relocated? 53. Sketch the three-dimensional figure formed by folding the net below into a solid. y R (2, 2) x A (–2, –1) B (2, –1) 54. Sketch... |
to your portfolio. Choose one homework assignment that demonstrates your best work in terms of completeness, correctness, and neatness. Add it (or a copy of it) to your portfolio. 92 CHAPTER 1 Introducing Geometry CHAPTER 2 Reasoning in Geometry That which an artist makes is a mirror image of what he sees around him. ... |
is likely to happen in the future. When you use inductive reasoning to make a generalization, the generalization is called a conjecture. Consider the following example from science. 94 CHAPTER 2 Reasoning in Geometry EXAMPLE A Solution A scientist dips a platinum wire into a solution containing salt (sodium chloride),... |
conjecture is difficult to find because the data collected are unorganized or the observer is mistaking coincidence with cause and effect. Good use of inductive reasoning depends on the quantity and quality of data. Sometimes not enough information or data have been collected to make a proper conjecture. For example, ... |
next five terms of the sequence in Exercise 18. 17. 3n 2 18. 1, 3, 6, 10,..., n(n 2 1),... 19. Now it’s your turn. Generate the first five terms of a sequence. Give the sequence to a member of your family or to a friend and ask him or her to find the next two terms in the sequence. Can he or she find your pattern? 20.... |
38. If two lines intersect to form a right angle, then they are?. For Exercises 39–42, sketch and label the figure. 39. Pentagon GIANT with diagonal AG parallel to side NT 40. A quadrilateral that has reflectional symmetry but not rotational symmetry 41. A prism with a hexagonal base 42. A counterexample to show that ... |
the way things come clear. All of a sudden. And then you realize how obvious they’ve been all along. MADELEINE L’ENGLE The success of an attorney’s case depends on the jury accepting the evidence as true and following the steps in her deductive reasoning. You use deductive reasoning in algebra. When you provide a reas... |
angle, or m 1 m. If m 180°, then 1 1 (180), 2 2 2 so 1 m 90°. The two angles are each less 2 than 90°, so they are acute. m1_ 2 m m1_ 2 Science Here is an example of inductive reasoning, supported by deductive reasoning. El Niño is the warming of water in the tropical Pacific Ocean, which produces unusual weather cond... |
used both inductive and deductive reasoning to convince yourself of the overlapping segments property. You will use a similar process in the next lesson to discover and prove the overlapping angles property in Exercise 17. Good use of deductive reasoning depends on the quality of the argument. Just like the saying, “A... |
both pairs of opposite sides parallel. What conclusion can you make? What type of reasoning did you use? 7. Use the overlapping segments property to complete each statement. B C D A a. If AB 3, then CD?. b. If AC 10, then BD?. c. If BC 4 and CD 3, then AC?. 8. In Example B of this lesson you discovered through inducti... |
ive reasoning was used incorrectly. Write a paragraph or two describing what happened and explaining why you think it was an incorrect use of inductive reasoning. 104 CHAPTER 2 Reasoning in Geometry Match each term in Exercises 20–29 with one of the figures A–O. 20. Kite 22. Trapezoid 24. Pair of complementary angles 2... |
calculate the 200th term. The rule that gives the nth term for a sequence is called the function rule. Let’s see how the constant difference can help you find the function rule for some sequences. DRABBLE reprinted by permission of United Feature Syndicate, Inc. Investigation Finding the Rule Step 1 Copy and complete ... |
at an example of how to find the 200th term in a geometric pattern. LESSON 2.3 Finding the nth Term 107 EXAMPLE B If you place 200 points on a line, into how many non-overlapping rays and segments does it divide the line? Solution Wait! don’t start placing 200 points on a line. You need to find a rule that relates the... |
5n 12 … n 5 f(n) 10 Term number 5 –10 –15 –20 108 CHAPTER 2 Reasoning in Geometry EXERCISES For Exercises 1–3, find the function rule f(n) for each sequence. Then find the 20th term in the sequence. 1. 2. 3. n f (n) n f (n) 1 3 1 1 2 9 3 15 4 21 5 27 6 … 33 … 3 2 … 2 5 8 11 14 … 5 4 6 1 n f (n) 4 2 4 3 12 4 20 5 28 6 ... |
that every triangle he drew turned out to have two sides congruent. José conjectures: “Look, Márisol, all triangles are isosceles.” How should Márisol respond? 15. A midpoint divides a segment into two congruent segments. Point M divides segment AY into two congruent segments AM and MY. What conclusion can you make? W... |
you think the answer would be? Choose a topic and a relationship to explore. You can use data from the census (such as age and income), or data you collect yourself (such as number of ice cubes in a. glass and melting time). For more sources and ideas, go to www.keymath.com/DG Collect pairs of data points. Use Fathom ... |
terms? If so, how confident are you of your generalization? To collect more data, you can ask more classmates to join your group. You can see, however, that acting out a problem sometimes has its practical limitations. That’s where you can use mathematical models. Step 3 Model the problem by using points to represent ... |
represents the number of points. What does the smaller of the two numbers in the numerator represent? Why do we divide by 2? Step 8 Write a function rule. How many handshakes were there at the party? 114 CHAPTER 2 Reasoning in Geometry The numbers in the pattern in the previous investigation are called the triangular ... |
order to have telephone service, how many lines are necessary? Is that practical? Sketch and describe two models: first, model the situation in which direct lines connect every house to every other house and, second, model a more practical alternative. 9. If each team in a ten-team league plays each of the other teams... |
, and manufacturing. To learn about new advances www.keymath.com/DG in organic chemistry, go to. IMPROVING YOUR VISUAL THINKING SKILLS Pentominoes II In Pentominoes I, you found the 12 pentominoes. Which of the 12 pentominoes can you cut along the edges and fold into a box without a lid? Here is an example. LESSON 2.4 ... |
can you be sure? As you do the next few steps, see if you can find the reason why some networks are impossible to travel. Step 2 Step 3 Draw the River Pregel and the two islands shown on the first page of this exploration. Draw an eighth bridge so that you can travel over all the bridges exactly once if you start at p... |
been given a name and a number. Start a list of them in your notebook. The Linear Pair Conjecture (C-1) and the Vertical Angles Conjecture (C-2) should be the first entries on your list. Make a sketch for each conjecture. 120 CHAPTER 2 Reasoning in Geometry In the previous investigation you discovered the relationship... |
the algebraic steps: m2 m3 180° m1 m2 180° m1 m2 m2 m3 thus m1 m3 therefore 1 3 This type of logical explanation, written as a paragraph, is called a paragraph proof. Now consider another idea. You discovered the Vertical Angles Conjecture: If two angles are vertical angles, then they are congruent. Does that also mea... |
sketch, then complete the following conjecture: If two angles are both congruent and supplementary, then?. 11. Using algebra, write a paragraph proof that explains why the conjecture from Exercise 10 is true. 12. Technology Use geometry software to construct two intersecting lines. Measure a pair of vertical angles. U... |
?? In other words, if there are n carbon atoms (C), how many 124 CHAPTER 2 Reasoning in Geometry 22. If the pattern of rectangles continues, what is the rule for the perimeter of the nth rectangle, and what is the perimeter of the 200th rectangle? Perimeter in a rectangular pattern Rectangle Perimeter of rectangle 1 10... |
see the other pair of alternate exterior angles? When parallel lines are cut by a transversal, there is a special relationship among the angles. Let’s investigate. You will need ● lined paper or a straightedge ● patty paper ● a protractor Step 1 Investigation 1 Which Angles Are Congruent? Using the lines on your paper... |
, is congruent. Will the lines be parallel? Is it possible for the angles to be congruent but for the lines not to be parallel? Investigation 2 Is the Converse True? You will need ● lined paper or a straightedge ● patty paper ● a protractor Step 1 Draw two intersecting lines on your paper. Copy these lines onto a piece... |
alternate interior angles. Therefore, if the corresponding angles are congruent, then the alternate interior angles are congruent. Solution It helps to visualize each statement and to mark all congruences you know on your paper. EXERCISES k m 3 1 2 Here are the algebraic steps: 2 1 3 1 So 2 3 Use your new conjectures ... |
middle tube is perpendicular to the top and bottom tubes. What are the measures of the incoming and outgoing angles formed by the light rays and the mirrors in this periscope? Are the surfaces of the mirrors parallel? How do you know? Review 12. What type (or types) of triangle has one or more lines of symmetry? 13. W... |
0 and maximums of 7. 2. The design consists of the graphs of seven lines. 3. The equation for one of the lines is y 1 x 1. 7 4. There’s a simple pattern in the slopes and y-intercepts of the lines. You’re on your own from here. Experiment! Then create a line design of your own and write the equations for it. 132 CHAPT... |
SKILLS 2 ● USING YO The slope is positive when the line goes up from left to right. The slope is negative when the line goes down from left to right. When is the slope 0? What is the slope of a vertical line? EXERCISES In Exercises 1–3 find the slope of the line through the given points. 1. (16, 0) and (12, 8) 2. (3, ... |
’s Sketchpad® can repeat a recursive rule on a figure using a command called Iterate. Using Iterate, you can create the initial stages of fascinating geometric figures called fractals. Fractals have self-similarity, meaning that if you zoom in on a part of the figure, it looks like the whole. A true fractal would need ... |
similarity. Notice that the fractal’s property of self-similarity does not change as you drag the vertices. Step 2 What happens to the number of shaded triangles in successive stages of the Sierpin´ski triangle? Decrease your construction to Stage 1 and investigate. How many triangles would be shaded in a Stage n Sierp... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.