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parallelogram. 2. If consecutive angles of a quadrilateral are supplementary, then the quadrilateral is a parallelogram. 3. If the diagonals of a quadrilateral are congruent, then the quadrilateral is a rectangle. 4. Two exterior angles of an obtuse triangle are obtuse. 5. The opposite angles of a quadrilateral inscribed within a circle are congruent. 6. The diagonals of a trapezoid bisect each other. 7. The midpoint of the hypotenuse of a right triangle is equidistant from all three vertices. In Exercises 8–12, complete each statement. 8. A tangent is? to the radius drawn to the point of tangency. 9. Tangent segments from a point to a circle are?. 10. The perpendicular bisector of a chord passes through?. 11. The three midsegments of a triangle divide the triangle into?. 12. A lemma is?. 13. Restate this conjecture as a conditional: The segment joining the midpoints of the diagonals of a trapezoid is parallel to the bases. CHAPTER 13 REVIEW 721 EW ● CHAPTER 13 REVIEW ● CHAPTER 13 REVIEW ● CHAPTER 13 REVIEW ● CHAPTER 13 REVIEW ● CH 14. Sometimes a proof requires a construction. If you need an angle bisector in a proof, what postulate allows you to construct one? 15. If an altitude is needed in a proof, what postulate allows you to construct one? 16. Describe the procedure for an indirect proof. 17. a. What point is this anti-smoking poster trying to make? b. Write an indirect argument to support your answer to part a. In Exercises 18–23, identify each statement as true or false. If true, prove it. If false, give a counterexample or explain why it is false. 18. If the diagonals of a parallelogram bisect the angles, then the parallelogram is a square. 19. The angle bisectors of one pair of base angles of an isosceles trapezoid are perpendicular. 20. The perpendicular bisectors to the congruent sides of an isosceles trapezoid are perpendicular. 21. The segment joining the feet of the altitudes on the two congruent sides of an isosceles triangle is parallel to the third side |
. 22. The diagonals of a rhombus are perpendicular. 23. The bisectors of a pair of opposite angles of a parallelogram are parallel. In Exercises 24–27, devise a plan and write a proof of each conjecture. 24. Refer to the figure at right. Given: Circle O with chords PN, ET, NA, TP, AE Show: mP mE mN mT mA 180° 25. If a triangle is a right triangle, then it has at least one angle whose measure is less than or equal to 45°. 26. Prove the Triangle Midsegment Conjecture. 27. Prove the Trapezoid Midsegment Conjecture. T A O N P E In Exercises 28–30, use construction tools or geometry software to perform each miniinvestigation. Then make a conjecture and prove it. 28. Mini-Investigation Construct a rectangle. Construct the midpoint of each side. Connect the four midpoints to form another quadrilateral. a. What do you observe about the quadrilateral formed? From a previous theorem, you already know that the quadrilateral is a parallelogram. State a conjecture about the type of parallelogram formed. b. Prove your conjecture. 722 CHAPTER 13 Geometry as a Mathematical System ● CHAPTER 13 REVIEW ● CHAPTER 13 REVIEW ● CHAPTER 13 REVIEW ● CHAPTER 13 REVIEW ● CHAP 29. Mini-Investigation Construct a rhombus. Construct the midpoint of each side. Connect the four midpoints to form another quadrilateral. a. You know that the quadrilateral is a parallelogram, but what type of parallelogram is it? State a conjecture about the parallelogram formed by connecting the midpoints of a rhombus. b. Prove your conjecture. 30. Mini-Investigation Construct a kite. Construct the midpoint of each side. Connect the four midpoints to form another quadrilateral. a. State a conjecture about the parallelogram formed by connecting the midpoints of a kite. b. Prove your conjecture. 31. Prove this theorem: If two chords intersect in a circle, the product of the segment lengths on one chord is equal to the product of the segment lengths on the other chord. Assessing What You’ve Learned WRITE IN YOUR JOURNAL How does the deductive system in geometry compare to the |
underlying organization in your study of science, history, and language? UPDATE YOUR PORTFOLIO Choose a project or a challenging proof you did in this chapter to add to your portfolio. ORGANIZE YOUR NOTEBOOK Review your notebook to be sure it’s complete and well organized. Be sure you have all the theorems on your theorem list. Write a one-page summary of Chapter 13. PERFORMANCE ASSESSMENT While a classmate, friend, family member, or teacher observes, demonstrate how to prove one or more of the theorems proved in this chapter. Explain what you’re doing at each step. GIVE A PRESENTATION Give a presentation on a puzzle, exercise, or project from this chapter. Work with your group, or try presenting on your own. CHAPTER 13 REVIEW 723 Hints for Selected Exercises... there are no answers to the problems of life in the back of the book. SØREN KIERKEGAARD in the text. Instead of turning You will find hints below for exercises that are marked with an to a hint before you’ve tried to solve a problem on your own, make a serious effort to solve the problem without help. But if you need additional help to solve a problem, this is the place to look. CHAPTER 0 • CHAPTER CHAPTER 0 • CHAPTER 0 LESSON 0.1 7. Here’s the title of the sculpture. Early morning calm knotweed stalks pushed into lake bottom made complete by their own reflection Derwent Water, Cumbria, 20 February & 8–9 March, 1988 LESSON 0.2 3. Design 1 Do one-half of the Astrid four times. Design 2 Connect the midpoints of the sides of an equilateral triangle with line segments and leave the middle triangle empty. Repeat the process on the three other triangles. Then, repeat the rule again. Design 3 On each side of an equilateral hexagon, mark the point that is one-third of the length of the side. Connect the points and repeat the process. LESSON 0.3 2. Use isometric dot paper, or draw an equilateral equiangular hexagon with sides of length 2. Each vertex of the hexagon will be the center of a circle with radius 1. Fit the seventh small circle inside the other six. The large circle has the same center as the seventh small circle and has radius 3. LESSON |
0.6 5. Draw two identical squares, one rotated 1 8 turn, or 45°, from the other. Where is the center of each arc located? CHAPTER 1 • CHAPTER CHAPTER 1 • CHAPTER 1 LESSON 1.1 3. Because a line is infinitely long in two directions, it doesn’t matter where the point used to name the line lies on the line. There are three possible ways to name the line if you don’t count switching the letters as two different ways. 21. Because a ray is infinitely long in one direction, it doesn’t matter which point you use for the second letter as long as it’s not the endpoint. LESSON 1.2 13. Find mCQA and mBQA and subtract. 26. Don’t forget that at half past the hour, the hour hand will be halfway between the 3 and the 4. 360°. Subtract the sum of 36. A 1 rotation 1 4 4 15° and 21° from that result. LESSON 1.3 15. Do not limit your thinking to just two dimensions. 24. The measure of the incoming angle equals the measure of the outgoing angle (just as in pool). 25. Use trial and error. LESSON 1.4 18. The order of the letters matters. 31. Draw your diagram on patty paper or tracing paper. Test your diagram by folding your paper along the line of reflection to see if the two halves coincide. Your diagram should have only one pair of parallel sides HINTS FOR SELECTED EXERCISES 725 34. Label the width of each small rectangle as w and its length as l. Therefore the perimeter of the large rectangle is 5w 4l 198 cm. 38. Only one of these is impossible. LESSON 1.5 24. There are four possible locations for R. The slope of CL 1. Therefore the slope 5 5 5 or of the perpendicular segment. 1 1 31. Locate the midpoint of each rod. Draw the segment that contains all the midpoints. This segment is called the median of the triangle. LESSON 1.6 19. This ordered pair rule tells you to double the abscissa and double the ordinate. (abscissa, ordinate) LESSON 1.7 3. Make a large graph (Quadrant I only). Label the vertical axis “feet” and the horizontal axis “days |
.” Or try a number line. 6. The vertical distance from the top of the pole to the lowest point of the cable is 15 feet. Compare that distance with the length of the cable. 7. Draw a diagram. Draw two points, A and B, on your paper. Locate a point that appears to be equally spaced from points A and B. The midpoint of AB is only one such point; find others. Connect the points into a line. For points in space, picture a plane between the two points. 11. Copy trapezoid ABCD onto patty paper or tracing paper. Rotate the tracing paper 90°, or 1 4 turn, counterclockwise. Point A on the tracing paper should coincide with point A on the diagram in the book. 15. The number of hexagons is increasing by one each time, but the perimeter is increasing by four. 34. Because PA, PB, QA, and QB are all radii, quadrilateral PAQB is a rhombus. AB and PQ are the diagonals of rhombus PAQB. LESSON 1.8 8. The biggest face is 3 m by 4 m. Your diagram will look similar to the diagram for Step 4 on page 80, except that the shortest segment will be vertical. 9. The number of boxes that will fit in the solid equals the volume, which is found by l × w × h. 18. Cut out a rectangle like the one shown and tape it to your pencil. Rotate your pencil to see what shape the rotating rectangle forms. 20. Imagine slicing an orange in half. What shape is revealed? 24. Do not limit your thinking to two dimensions. This situation can be modeled by using three pencils to represent the three lines. CHAPTER 1 REVIEW 37. Here is one possible method. Draw a circle and one diameter. Draw another diameter perpendicular to the first. Draw two more diameters so that eight 45° angles are formed. Draw the regular octagon formed by the endpoints of the diameters. 45. A clock forms 12 central angles that each measure 30° 3 0°. The angle formed by the 6 1 2 hands is 31 of those central angles. 2 54. Cut out a semicircle and tape it to your pencil. Rotate your pencil to see what shape the rotating semicircle forms. 57. Rotate your book so that the red line is vertical. CHAPTER 2 • CHAPTER CHAPTER 2 • CH |
APTER 2 LESSON 2.1 1. Conjectures are statements that generalize from a number of instances to “all.” Therefore, Stony is saying “All?.” 4. Change all fractions to the same denominator. 7. 1 1 2, 1 2 3, 2 3 5, 3 5 8 726 HINTS FOR SELECTED EXERCISES 8. 12, 22, 32, 42,... 13. Add another row with one more rectangle. 14. Each segment branches off into two segments. 15. Connect the midpoints of the sides of the triangle. Make the triangle formed in the middle white. Connect the midpoints of the sides of the shaded triangles. Make the triangles formed in the middle white. 17. Substitute 1 for n and evaluate the expression to find the first term. Substitute 2 for n to find the second term, and so on. 21. For example, “I learned by trial and error that you turn wood screws clockwise to screw them into wood and counterclockwise to remove them.” 26. Imagine folding the square up and “wrapping” the two rectangles and the other triangle around the square. 27. Turn your book so that the red line is vertical. Imagine rotating the figure so that the part jutting out is facing back to the right. 28. Cut out a quarter-circle and tape it to your pencil. Rotate your pencil to see the figure formed. 42. Remember that a kite is a quadrilateral with two pairs of consecutive, congruent sides. One of the diagonals does bisect the angles of the kite, the other does not. LESSON 2.2 8. What is the smallest possible size for an obtuse angle? 9. Compare how many 1’s there are in the numbers that are multiplied with the middle digit of the answer; then compare both quantities with the row number. 12. Look for a constant difference among terms. 16. The number of rows increases by one, and the number of columns increases by one. 17. The number of rows increases by two, and the number of columns increases by one. LESSON 2.3 1. Look for a constant difference, then adjust the rule for the first term. 4. See below. 7. Draw the polygons and all possible diagonals from one vertex. Fill in the table and look for a pattern. |
You should be able to see the pattern by the time you get to a hexagon. LESSON 2.4 4. Compare this exercise with the Investigation Party Handshakes, and with Exercise 3. What change can you make to each of those functions to fit this pattern? 5. This is like Exercise 4 except that you add the number of sides to the number of diagonals. 6. In other words, each time a new line is drawn, it passes through all the others. This also gives the maximum number of intersections. 7. Exercises 5 and 6 have the same rule. Every term in the sequence for Exercise 4 is n less than the corresponding term in the sequences for Exercises 5 and 6. This is because in Exercise 5 you count the sides of the polygons. 9. Let a point represent each team, and let the segments connecting the points represent one game played between them. What do you then have to multiply this answer by? 10. Use the model from Exercise 5. 14. The tricky part to this problem is that points A and B could be on the same side of point E. 4. (Lesson 2.3) Figure number Number of tiles HINTS FOR SELECTED EXERCISES 727 LESSON 2.5 20. Proceed in an organized way. Number of 1-by-1 squares? Number of 2-by-2 squares? Number of 3-by-3 squares? Number of 4-by-4 squares? Then add. 21. You can add the first two function rules to find the third function rule. CHAPTER 2 REVIEW 5. Try the alphabet backward and the powers of 2 forward. 7. For the pattern of the letters, number each letter of the alphabet and find the pattern of their differences. 9. Here is how to begin: f(1) 211 20 1 f(2) 221 21 2 11. The diagrams are alternating net and solid. The number of sides of the base increases by one. 14. Refer to Lesson 2.4, the Investigation Party Handshakes, discussion of triangular numbers. 15. Exercises 14 and 15 are closely related. Let’s examine the rule for Exercise 14 We want 1 to be the first term, not 0. Now look at the pattern for the rest of the terms. The underlined numbers are the n, and the numbers in front of the n are one higher. Here is how to begin: a 60° because of the |
Vertical Angles Conjecture. c 120° because of the Linear Pair Conjecture. 23. Refer to Lesson 2.4, Exercise 5. 24. Refer to Lesson 2.4, Exercise 6. 25. Refer to Lesson 2.4, Exercise 5, but subtract the number of couples from each term because the couples don’t shake hands. 27. Refer to Lesson 2.4, Exercise 4. Then use “guess and check.” LESSON 2.6 4. Because quadrilateral TUNA is a parallelogram, TU AN and TA UN. Form NA by extending side NA. Place a point Q on NA, to the left of A. T QAT by the Alternate Interior Angles Conjecture. N QAT by the Corresponding Angles Conjecture. Because T and N are both congruent to QAT, N T. 6. E D C z m A 67° B n DAB and EDA are congruent by the Alternate Interior Angles Conjecture. EDA and CDA are a linear pair. Now every angle in quadrilateral ABCD is known except the one labeled z. The sum of the angles of a quadrilateral is 360°. 7. Measures a, b, c, and d are all related by parallel lines. Measures e, f, g, h, i, j, k, and s are also all related by parallel lines. 13. Squares, rectangles, rhombuses, and kites are eliminated because they have reflectional symmetry. 16. Graph the original triangle and the new triangle on separate graphs. Cut one out and lay it on top of the other to see if they are congruent. 728 HINTS FOR SELECTED EXERCISES 17. See below. 18. 1 1 12 1 3 4 22 1 3 5 9 32 1 3 5 7 16 42 And so on... 20. How many vertical interior segments are there? How many horizontal? 23. Refer to Lesson 2.4, Exercise 6. Then use “guess and check.” CHAPTER 3 • CHAPTER CHAPTER 3 • CHAPTER 3 LESSON 3.1 2. Copy the first segment onto a ray. Copy the second segment immediately after the first. 10. You duplicated a triangle in Exercise 7. You can think of the quadrilateral as two triangles stuck together ( |
they meet at the diagonal). 14. Fold the paper so that the two congruent sides of the triangle coincide. LESSON 3.2 2. Bisect, then bisect again. 17. (Chapter 2 Review) 3. Construct one pair of intersecting arcs, then change your compass setting to construct a second pair of intersecting arcs on the same side of the line segment as the first pair. 4. Bisect CD to get the length 1 CD. Subtract this 2 length from 2AB. 5. The average is the sum of the two lengths divided by the number of segments (two). Construct a segment of length AB CD. Bisect the segment to get the average length. Or take half of each, then add them. 8. Construct the median from the vertex to the midpoint. LESSON 3.3 3. Construct the perpendicular through point B to TO. 4. Does your method from Investigation 1 still work? Can you modify it? 5. Construct right angles at Q and R. 13. Look at two columns as a “group.” 1st rectangle is 2 groups of 1 2nd rectangle is 3 groups of 3 3rd rectangle is 4 groups of 5 4th rectangle is 5 groups of 7... nth rectangle is n 1 groups of 2n 10 4 6 2 6 12 2 8 20 2 30 10 The differences between terms aren’t constant, so it’s not a linear pattern, but the second set of differences is a constant. Sequences of this type have two linear factors, as the table below shows, and are called quadratic sequences. n Term Factors 12 3 × 4 20 4 × 5 30 HINTS FOR SELECTED EXERCISES 729 17. Each angle of a regular pentagon is 108°. 180° 72° _________ 2 54° 360° _____ 5 72° 54° 72 54° 54° 108° (Or, divide a circle into five congruent arcs, and join the endpoints.) LESSON 3.4 14. Use your protractor and measure off eight rays 45° apart about a point. Use your compass and swing a circle at the point of intersection. 15. Construct two lines perpendicular to each other, and then bisect the right angles. 16. Because the angles of a triangle add to 180°, the problem could be restated as “Draw a second triangle with a 40° angle, an 80° angle, and a |
side between the given angles measuring 8 cm.” LESSON 3.5 3. If the perimeter is z, then each side has length 1 z. Construct the perpendicular bisector of z to 4 get 1 z. Construct the perpendicular bisector of 1 z 2 2 to get 1 z. 4 10. According to the Perpendicular Bisector Conjecture, if a point is on the perpendicular bisector, then it is equidistant from the endpoints of the segment (which in this case represent fire stations). Therefore, if a point is on one side of the perpendicular bisector, it is closer to the fire station on that side of the perpendicular bisector. LESSON 3.6 1. Construct a segment, MS. Draw an arc with radius AS from point S and an arc with radius MA from point M. Connect the point at which the arcs intersect to points M and S to form your triangle. 730 HINTS FOR SELECTED EXERCISES 5. Duplicate A and AB on one side of A. Open the compass to length BC. If you put the compass point at point B, you’ll find two possible locations to mark arcs for point C. C A B A C B 6. y x is the sum of the two equal sides. Find this length and bisect it to get the length of the other two legs of your triangle. LESSON 3.7 6. Find the incenter. 7. Find the circumcenter. 8. Draw a slightly larger circle on patty paper and try to fit it inside the triangle. 9. Draw a slightly smaller circle on patty paper and try to fit it outside the triangle. 16. Start by finding points whose coordinates add to 9, such as (3, 6) and (7, 2). Try writing an equation and graphing it. 17. One way is to construct the incenter by bisecting the two given angles. Then find two points on the unfinished sides, equidistant from the incenter and in the same direction (both closer to the missing point.) Now find a point equidistant from those two points. Draw the missing angle bisector through that point and the incenter. LESSON 3.8 2. If CM 16, then UM 1 (16) 8. If TS 21, 2 then SM 1 (21) 7. 3 8. A quadrilateral can be divided into two triangles in two different ways. How can you use the cent |
roids of these triangles to find the centroid? 15. Construct the altitudes for the two other vertices. From the point where the two altitudes meet, construct a line perpendicular to the southern boundary of the triangle. 16. How many people does each person greet? Don’t count any greeting twice. There are 60 people. If everyone greets each other, the 59. But dorm number of greetings would be 60 2 members aren’t greeting their guests, so the 59 40. number of greetings would be 60 2 CHAPTER 3 REVIEW 28. Draw a long segment and use your compass to add y plus y plus x, bisect z, then subtract it from the sum. CHAPTER 4 • CHAPTER CHAPTER 4 • CHAPTER 4 LESSON 4.1 4. Ignore the 100° angle and the line that intersects the larger triangle. Find the three interior angles of the triangle. Then find z. 6. The total measure of the three angles is 3 times 360° minus the sum of the interior angles of the triangle. 7. The sum of a linear pair of angles is 180°. So the total measure of the three angles is 3 times 180° minus the sum of the interior angles of the triangle. 8. You can find a by looking at the large triangle that has 40° and 71° as its other measures. You can find b because it forms a linear pair with the 133° angle. Continue on your own. 15. mA mB x 180° and mD mE y 180°. Then use substitution to prove x y. LESSON 4.2 1. mH mO 180° 22° and mH mO. 7. Notice that d e and d e e 66° 180°. Next, find the alternate interior angle to c. 8. Notice that all the triangles are isosceles! 21. Move each point right 5 units and down 3 units. LESSON 4.3 5. Find the value of the unmarked angle and use the Side-Angle Inequality Conjecture. 9. Use the Side-Angle Inequality Conjecture to find an inequality of sides for each triangle, then combine the inequalities. 11. Any side must be smaller than the sum of the other two sides. What must it be larger than? 13. Try using the Triangle Sum Conjecture. 14. Use the Triangle Exterior Angle Conjecture. 17. You need to show that x |
a b. From the Triangle Sum Conjecture, you know that a b c 180°. Also, BCA and BCD are a linear pair, so x c 180°. 21. All corresponding sides and angles are congruent. Can you see why? (And remember that the ordering of the points is important in correctly stating the answer.) LESSON 4.4 1. Rotate one triangle 180°. 2. The shared side is congruent to itself. 9. Match congruent sides. 12. Take a closer look. Are congruent parts corresponding? 15. UN YA 4, RA US 3, mA mU 90° LESSON 4.5 3. Flip one triangle over. 19. The sides do not have to be congruent. 12. The sides do not have to be congruent. LESSON 4.6 13. Make GK MP. 15. Find the slopes. 1. Don’t forget to mark the shared segment as congruent to itself. HINTS FOR SELECTED EXERCISES 731. Use CRN and WON. 4. Use ATI and GTS. 5. Draw UF. 7. Draw UT. 10. Count the lengths of the horizontal and vertical segments, and label the right angles congruent. 17. Make the included angles different. LESSON 4.7 7. Use ABD and CBD. 8. Don’t forget about the shared side and the Side-Angle Inequality Conjecture. 10. When you look at the larger triangles, ignore the marks for OS RS. When you look at the smaller triangles, ignore the right-angle mark and the given statement PO PR. 11. First, mark the vertical angles. In ADM, the side is included between the two angles. In CRM, the side is not included between the two angles. 14. Review incenter, circumcenter, orthocenter, and centroid. 17. See if your teacher has 14 cubes. Build the figure shown, take away the indicated blocks, and draw what is left. LESSON 4.8 1. AB BC AC 48 AD 1 AB 2 8. Use the vertex angle bisector as your auxiliary line segment. 12. At 3:15 the hands have not yet crossed each other. At 3:20 the hands have already crossed each other, because the minute hand is on the 4 but the hour hand is only one- |
third of its way from the 3 toward the 4. So the hands overlap sometime between 3:15 and 3:20. 14. Make a table and look for a pattern. 17. How many H’s branch off each C? 732 HINTS FOR SELECTED EXERCISES CHAPTER 4 REVIEW 19. Look for alternate interior angles. 21. Not enough information is given. The two angles at point H may look the same, but you just don’t know. 23. There are actually two isosceles triangles in the figure, and there are three possible answers. It may help to redraw the triangles so that they don’t overlap. 25. Calculate the missing angle measure. 32. Look for congruent triangles. Then look for congruent angles to show that lines are parallel. 35. Vertical angles are congruent. CHAPTER 5 • CHAPTER CHAPTER 5 • CHAPTER 5 LESSON 5.1 2. All angles are equal in measure. 6. d 44° 30° 180° 7. 3g 117° 108° 540° 13. (n 2) 180° 2700° ) 180° 156° 14. (n 2 n 17. Use the Triangle Sum Conjecture and angle addition. LESSON 5.2 6. First, find the measure of an angle of the equiangular heptagon. Then, find c by the Linear Pair Conjecture. 0° 10. 24° 36 n 12. An obtuse angle measures greater than 90°, and the sum of exterior angles of a polygon is always 360°. 15. Look at RAC and DCA. 16. Draw AT. LESSON 5.3 11. Construct I and W at the ends of WI. Construct IS. Construct a line through point S parallel to WI. 13. The nonshared endpoints of the consecutive congruent segments are equidistant from the shared endpoint. So the shared endpoint is on the perpendicular bisector of one diagonal. 16. Look at AFG and BEH. LESSON 5.4 2. PO 1 RA 2 18. Construct B, then bisect it. Mark off the length of diagonal BK on the angle bisector. Then construct the perpendicular bisector of BK. 23. The diagonal of a rhombus bisects the angle. 24. Construct a rhombus with your segment as the diagonal. The other diagonal will be |
the perpendicular bisector. 31. Use the Quadrilateral Sum Conjecture to prove the measure of each angle is 90°. If the consecutive interior angles are supplementary, then the lines are parallel. LESSON 5.7 4. Use the Three Midsegments Conjecture. 1. If you start from square 100 and work backward, the problem becomes much easier. 9. Draw a diagonal of the original quadrilateral. Note that it’s parallel to two other segments. 7. Look at YIO and OGY. LESSON 5.5 VF and NI 1 4. VN 1 EI 2 2 8. With the midpoint of the longer diagonal as center and using the length of half the shorter diagonal as radius, construct a circle. 10. Complete the parallelogram with the given vectors as sides. The resultant vector is the diagonal of the parallelogram. Refer to the diagram right before the Exercises. 11. PR a. Therefore MA a.? a b. Solve for?. The height of A is c. Therefore the height of M is c. LESSON 5.6 1. Consider the parallelogram below. 12. P and PEA are supplementary. 17. The diagonals of a square are equal in length and are perpendicular bisectors of each other. 8. Break up the rectangle into four triangles (EAR, etc.), and show that they are congruent triangles. 12. Imagine what happens to the rectangles when you pull them at their vertices. 13. Calculate the measures of the angles of the regular polygons. Remember that there are 360° around any point. 14. Look at the alternate interior angles. 15. You miss 5 minutes out of 15 minutes. full. It will be 2 2 8 full 16. The container is 3 3 1 2 no matter which face it rests on. CHAPTER 5 REVIEW 20. Refer to the figure at the end of the Investigation Four Parallelogram Properties for a similar example. Let 1 cm 100 km be your scale. 23. Construct LP and copy L. Mark off LN. At point N, construct a line parallel to LP HINTS FOR SELECTED EXERCISES 733 CHAPTER 6 • CHAPTER CHAPTER 6 • CHAPTER 6 LESSON 6.1 4. y y y 72° 360° 18. Calculate the slope and midpoint of AB. Recall that slopes of perpendicular lines are negative reciproc |
als. 20. NN 15° 117° 9 km 6 km E E LESSON 6.2 1. 130° 90° w 90° 360° 2. x x 70° 180° 5. CP PA AO OR, CT TD DS SR 8. From the Tangent Conjecture, you know that the tangent is perpendicular to the radius at the point of tangency. 15. Draw a diameter. Then bisect it repeatedly to find the centers of the circles. 16. Look at the angles in the quadrilateral formed. 23. Draw the triangle formed by the three light switches. The center of the circumscribed circle would be equidistant from the three points. 24. Make a list of the powers of 3, beginning with 3° 1. Look for a pattern in the units digit. 25. Use a protractor and a centimeter ruler to make a careful drawing. Let 1 cm represent 1 mile. LESSON 6.3 3. c 120° 2(95°) 4. Draw in the radius to the tangent to form a right triangle. 734 HINTS FOR SELECTED EXERCISES 14. The measure of each of the five angles is half the measure of its intercepted arc. But the five arcs add up to the complete circle (360°). (70°), b 1 15. a 1 (80°), y a b 2 2 80° a y b 70° 19. Draw the altitude to the point where the side of the triangle (extended if necessary) intersects the circle. 20. One possible location for the camera to get all students in the photo Students lined up for the photo 23. Show congruent right triangles inside congruent isosceles triangles. 24. Start with an equilateral triangle whose vertices are the centers of the three congruent circles. Then locate the incenter/circumcenter/ orthocenter/centroid to find the center of the larger circle. LESSON 6.4 4. Note that mYLI 1. Use angle addition, arc addition, and Case 1. mYCI 5. 1 2 by AIA, and 1 and 2 are inscribed angles. 360°. 6. Apply the Cyclic Quadrilateral Conjecture. 7. Draw one diagonal and use the Inscribed Angle Conjecture. 9. Think about the pair of angles that form a linear pair and the isosceles triangle. 10. Number the points in the grid |
1–9. Make a list of all the possible combinations of three numbers. Do this in a logical manner. Order doesn’t matter, so the list beginning with 2 will be shorter than the list beginning with 1. The 3 list will be shorter than the 2 list, and so on. See how many of the possibilities are collinear, and divide that by the total number of possibilities. 11. Make an orderly list. Here is a beginning: RA to AL to LG RA to AN to NG LESSON 6.5 9. C 2r, 44 2(3.14)r 12. The diameter of the circle is 6 cm. 17. Use the Inscribed Angle Conjecture and the Triangle Exterior Angle Conjecture. LESSON 6.6 nce f e r e ce circ m u an t s 1. speed di (2000 6400) km 12 hours 8. Calculate the distance traveled in one revolution, or the circumference, at each radius. Multiply this by the rpm to get the distance traveled in 1 minute. Remember, your answer is in inches. You may want to divide your answer by 12 and then by 60 to change its units into feet per second. 10. Use the Inscribed Angle Conjecture and the Triangle Exterior Angle Conjecture. LESSON 6.7 1 0 3. 2 (24) 6 0 3 70° mAR 8. mAR A R m (2r) 40 0 36 ° 146° 360°, 9. The length of one lap is equal to (2 100) (2 20). The total distance covered in 6 minutes is 4 laps. 11. 1 (2r) 12 meters 9 15. The midsegment of a trapezoid is parallel to the bases, and the median to the base of an isosceles triangle is also the altitude. 16. The overlaid figure consists of two pairs of congruent equilateral triangles. The length of the side of the smaller pair is half the length of the side of the larger pair. All of the arcs use the lengths of the sides of the triangles as radii. 17. It is not 180°. What fraction of a complete cycle has the minute hand moved since 10:00? Hasn’t the little hand moved that same fraction of the way from 10:00 to 11:00? CHAPTER 6 REVIEW 5. Draw in the radius to the tangent to form a right triangle. |
8. See Lesson 6.5, Exercises 16 and 17. 10. The supplement of 88° is 92°. 1 (118° f ) 92°. 2 32 12. C d, 132 d, d 1 (54). is 1 0 0 ° 3 ° 0 6 14. To find the length of DC, first find the 60° 50°.. mDL degree measure of DL 2 13. Arc length of AB 29. Here is how to calculate 1 nautical mile near 7. 5 63 a pole: 2 0 6 60 3 30. The locus of possible locations for Dmitri is a circle with radius 5 1100 ft 5500 ft, and the locus for Tara is a circle with radius 7 1100 ft 7700 ft. How many times can the two circles intersect? 31. The circumference of the table is 2 × 100. Calculate the diameter HINTS FOR SELECTED EXERCISES 735 CHAPTER 7 • CHAPTER CHAPTER 7 • CHAPTER 7 CHAPTER 8 • CHAPTER CHAPTER 8 • CHAPTER 8 LESSON 7.2 LESSON 8.1 2. All positive y’s become negative y’s; therefore, the figure is reflected over the x-axis. 7. Compare the ordered pairs for V and V, R and R, and Y and Y. 10. Factor 48 in two different ways. 19. Convert inches to fractions of a foot. LESSON 8.2 LESSON 7.3 8. 50 1 h(7 13) 2 7. Connect a pair of corresponding points with a segment. Construct two perpendiculars to the segment with half the distance between the two given figures between them. 16. Find the midpoint of the segment connecting the two points. Connect the midpoint to one of the endpoints with a curve. Copy the curve onto patty paper and rotate it about the midpoint. LESSON 7.4 13. 12. The area of the triangle can be calculated in three different ways, but each should give the (15)x 1 (5)y 1 same area: 1 (6)(9). 2 2 2 22. Refer to the diagram below. h b1 I II b2 Area of I b1h 1_ 2 b1 → h I II h b2 Area of II b2h 1_ 2 29. Draw the prism unfolded. P A T C M U B LESSON 8.3 18. Work backward. Reflect |
a point of the 8-ball over the S cushion. Then reflect this image over the N cushion. Aim at this second image. LESSON 7.5 2. Connect centers across the common side. 4. Total cost is $20/yd2 $20/9 ft2; Acarpet 17 27 (6 10 7 9); 1 yd 3 ft 8. First, find the area of all the vertical rectangles (walls). Notice that the area of the front and back triangles are the same. LESSON 7.6 LESSON 8.4 9. If you are still unsure, use patty paper to trace the steps in the examples (“Pegasus” and Monster Mix). LESSON 7.8 5. If you are still unsure, use patty paper to trace the four steps in the Escher Horseman example and the Escher Symmetry Drawing E108 example. 9. Construct a circle with radius 4 cm. Mark off six 4 cm chords around the circle. 10. Draw a regular pentagon circumscribed about a circle with radius 4 cm. Use your protractor to create five 72° angles from the center. Use your protractor to draw five tangent segments. 13. Find the area of the large hexagon and subtract from it the area of the small hexagon. Because they 736 HINTS FOR SELECTED EXERCISES are regular hexagons, the distance from the center to each vertex equals the length of each side. 15. Divide the quadrilateral into two triangles (A and B). Find the areas of the two triangles and add them. y A 1_ y x 5 2 (2, 6) y 2x 10 B x LESSON 8.5 15. Calculate the area of two circles, one with radius 3 cm and one with radius 6 cm. Compare the two areas. 16. a r a r To find the area of the six lateral faces, imagine unwrapping the six rectangles into one rectangle. The lateral area of this “unwrapped” rectangle is the height times the perimeter. 9. 4 C 8 C 4 8 9 Top and bottom 9 Outer surface Inner surface CHAPTER 8 REVIEW 37. Refer to Lesson 8.2 Project Maximizing Area. 39. r 43a. 3.2 r r r r Hexagon Dodecagon? 100-gon LESSON 8.6 6. The shaded area is equal to the area of |
the whole circle less the area of the smaller circle. x (102 82) 12. 10 36 0 hb1 18. A 1 b2 2 the midsegment is 1 b1 2 be rewritten A midsegment height.. Because the length of b2, the formula can LESSON 8.7 7. Use the formula for finding the area of a regular hexagon to find the area of each base. 12 5 43b. Total surface area area of octagon 8 area of small trapezoid 8 area of large trapezoid 8 area of square. CHAPTER 9 • CHAPTER CHAPTER 9 • CHAPTER 9 LESSON 9.1 6. 62 62 c2 11. The radius of the circle is the hypotenuse of the right triangle. 13. Let s represent the length of the side of the square. Then s2 s2 322. 15. Three consecutive integers can be written algebraically as n, n 1, and n 2. HINTS FOR SELECTED EXERCISES 737 17. Show that the area of the entire square (c2) is equal to the sum of the areas of the four right triangles and the area of the smaller square. b a b a c 16. Draw an altitude of the equilateral triangle to form two 30°-60°-90° triangles. 19. Construct an isosceles right triangle with legs of length a; construct an equilateral triangle with sides of length 2a and construct an altitude; and construct a right triangle with legs of length a2 and a3. 22. Make the rays that form the right angle into lines. OR: Draw an auxiliary line parallel to the other parallel lines through the vertex of the right angle. LESSON 9.2 LESSON 9.4 6. a2 b2 must exactly equal c2. 10. Drop a perpendicular from the ordered pair to the x-axis to form a right triangle. 12. Check the list of Pythagorean triples in the beginning of this lesson for a right triangle that has three consecutive even integers. 22. Because a radius is perpendicular to a tangent, mDCF 90°. Because all radii in a circle are congruent, DCE is isosceles. 1. The length of the hypotenuse is (36 x). Solve for x. x 36 x 24 d 3. Average speed ours 4 h LESSON 9.3 160 km d 2. b hypoten |
use 2 4. d 1 20, c d 3 2 7. Draw diagonal DB to form a right triangle on the base of the cube and another right triangle in the interior of the cube. 9. Divide by 2 for the length of the leg. 12. This is one way to show the relationship. Draw three 30°-60°-90° triangles with sides of lengths 1, 3, and 2. 62 32 27, so 33 27 738 HINTS FOR SELECTED EXERCISES Lost Wages 30 km/hr for 4 hrs Pecos Gulch 4. This is a two-step problem, so draw two right triangles. Find h, the height of the first triangle. The height of the second is 4 ft less than the height of the first. Then find x. h 25 h 4 25 x 7 7 x 7 5. 13 m 13 m 10 m 13 13 h 5 5 6. Find the apothem of the hexagon. 16. Use the Reflection Line Conjecture and special right triangles. LESSON 9.5 11. Use the distance formula to find the length of the radius. 12. This is the same as finding the length of the space diagonal of the rectangular prism. 14. In the 45°-45°-90° triangle, m 3 2; in the 30°-60°-90° triangle, m k3. 16. (x 8)2 402 x2 LESSON 9.6 1. Find mDOB using the Quadrilateral Sum Conjecture. 3. See below. 5. When OT and OA are drawn, they form a right triangle, with OA 15 (length of hypotenuse) and OT 12 (length of leg). 18 3 12 18 60° 60° 12 3 60° 60° 18 36 12 3 3 24 8. 10. 60° 30° 8 0 [2(9)] 4. is 14. The arc length of AC 0 6 3 Therefore, the circumference of the base of the cone is 4. From this you can determine the radius of the base. The radius of the sector (9) becomes the slant height (the distance from the tip of the cone to the circumference of the base). The radius of the base, the slant height, and the height of the cone form a right triangle. 19. Rearrange the equation: x2 2x 1 y2 100 (? )2 y2 100 CHAPTER 9 REVIEW 10. |
The diameter of the semicircle is the longer leg of a 7-? -25 right triangle. 12. Each half of the shaded area is equal to a quarter of a circle less the area of the isosceles right triangle. 15. (45 2)2 (60 2)2 d2 “Pay dirt” 45 km/hr for 2 hrs “No luck” d Camp 60 km/hr for 2 hrs 16. What will be the length of the diagonal if the shape is a rectangle? CHAPTER 10 • CHAPTER 10 CHAPTER 10 • CHAPTER LESSON 10.1 29. Think of a prism as a stack of thin copies of the bases. 31. 8 3 A 8 60° R T 8 3 H 8 60° 8 HINTS FOR SELECTED EXERCISES 739 LESSON 10.2 bhH 2. V BH 1 2 5. You have only 1 of a cylinder. 2 r2H V BH 1 2, or 1 9 0 6., of the cylinder is removed. 4 0 6 3 Therefore, you need to find 3 of the volume of 4 the whole cylinder. 7a. What is the difference between this prism and the one in Exercise 2? Does it make a difference in the formula for the volume? 9. Cutie pie! 26. SOTA is a square, so the diagonals, ST and OA, are congruent and are perpendicular bisectors of each other and SM OM 6. Use right triangle SMO to find OS, which also equals OP. Find PA with the equation PA OA OP. S A P 6 6 M O T LESSON 10.3 b2 hH b1 BH 1 3. V 1 2 3 3 Vcone 6. V Vcylinder BH 1 BH 3 10a. What is B, the area of the triangular base? 15. 8 6 6 740 HINTS FOR SELECTED EXERCISES 18. The swimming pool is a pentagonal prism resting on one of its lateral faces. The area of the pentagonal base can be found by dividing it into a rectangular region and a trapezoidal region. 30 ft 4 ft 4 ft 10 ft 13 ft LESSON 10.4 (20 28)h(36) 1 BH; 3168 1 4. V 1 2 3 3 Vmissing prism Vlarger prism 9. Vring 23(24)(2) 33 |
the triangle will trace the path of a circle. So, if you connect any vertex of the original figure to its corresponding vertex in the image, you will get a chord. Now, recall that the perpendicular bisector of a chord passes through the center of a circle. LESSON 11.2 3. It helps to rotate ARK so that you can see which sides correspond. T 36 42 DK A A m 24 R 8. 3 cm 2 cm 45 cm 30 cm d 20 cm 9. Think about what Juanita did in Exercise 5. 11. Draw the large and small triangles separately to label and see them more clearly. 13. 20 48 52 h y 48 x h 20 17. The golden ratio is How would you construct the length 5? How would you construct the length 5 1? 2. Let AB 2 units. 5 1 HINTS FOR SELECTED EXERCISES 741 LESSON 11.4 L C C S, I C C 3. I I SE E I PE E 2 9. 1 x x 10 1 5 12. You don’t know the length of the third side, but you do know the ratio of its two parts is 2. 3 h2 12. H 9 ; 2 5 Volume of large prism Volume of small prism h3 H 17. Volume of large warehouse 2.53 (Volume of small warehouse) 26. What kind of triangle is this? 27. A cross section is a section that is perpendicular to the axis 3a 2a 15 x 10 16. Bisect the angle between the sides of lengths 2x and 3x. 17. Use the AA Similarity Conjecture for this proof. One pair of corresponding angles will be a nonbisected pair. The other pair of corresponding angles will consist of one-half of each bisected angle. 20. a c 1, then c, then 26. Copy the diagram on your own paper, and connect the centers of two large circles and the center of the small circle to form a 45°-45°-90° triangle. Each radius of the larger circles will be s. Use the properties of special right triangles and 4 algebra to find the radius of the smaller circle. LESSON 11.5 2 1. 1 2 MSE Area ID O Z, then the ratio of the 3. If lengths of corresponding sides is 4. 5 7. m n Area 6m2 Area 6n2 742 HINTS FOR SELECTED EXERCISES LESSON 11. |
6 4 a 1. 4 2 0 12 c 60 0 3. 6 70 0 4 6. 2 3 6, then 1 3 4 4. If. 1 5 5 3 6 12. a b (12 3)2 (0 9)2. Once you 4 a have solved for a b, then. a 1 2 b x 12; x is the height of the small x 17. 16 1 0 missing cone. 12 cm 10 cm 16 cm CHAPTER 11 REVIEW 9. Divide KL into seven equal lengths. D B A 10. ABE ADC, A CD BE A E B 2 m 4 m 20 m r C D 19. Because you are concerned with ratios, it doesn’t make any difference what lengths you choose. So, for convenience, assign the square a side of length 2 before calculating the areas and volumes. CHAPTER 12 • CHAPTER 12 CHAPTER 12 • CHAPTER LESSON 12.1 7. The length of the side opposite A is s; the length of the side adjacent to A is r; the length of the hypotenuse is t. 10. Use your calculator to find sin1(0.5). 0 14. tan 30° 2 a b 21. Use sin 35° to find the length of the 8 5 h base, and then use cos 35° to find the height. 8 5 27. First, find the length of the radius of the circle and the length of the segment between the chord and the center of the circle. Use the Pythagorean Theorem to find the length of half of the chord. 4 cm 12 cm LESSON 12.3 3. Sketch a diagonal connecting the vertices of the unmeasured angles. Then find the area of the two triangles. 4. Divide the octagon into eight isosceles triangles. Then use trigonometry to find the area of each triangle. 79° 2° sin 5 5. sin 28 ° w a 13. sin 16° 8 1 b cos 16° 8 1 c tan 68° b c b 16° a 68° 18 m 16° LESSON 12.4 1. w2 362 412 2(36)(41)cos 49° 4. 422 342 362 2(34)(36)cos A 8. The smallest angle is opposite the shortest side. 10. One approach divides the triangle into two right triangles, where x a b. When you have the straight-line distance, compare that time against the detour. LESSON |
travel, 280–281, 337, 338, 340, 392, 423, 570, 644, 648, 649, 660 Alexander the Great, 18 algebra Activity: Dilation Creations, properties of, in proofs, 670–671, 578–580 Activity: Dinosaur Footprints and Other Shapes, 431–432 Activity: Elliptic Geometry, 719–720 Activity: Exploring Properties of Special Constructions, 696–697 Activity: Exploring Star Polygons, 264–265 Activity: Isometric and 712 See also Improving Your Algebra Skills; Using Your Algebra Skills Algeria, 567 Almagest (Ptolemy), 620 alternate exterior angles, 126–129 Alternate Exterior Angles Conjecture (AEA Conjecture), 127 alternate interior angles, 126–129, Orthographic Drawings, 541 681 Activity: It’s Elementary!, 552–553 Activity: Modeling the Platonic Solids, 528–530 Alternate Interior Angles Conjecture (AIA Conjecture), 127, 129 Alternate Interior Angles Theorem (AIA Theorem), 681 Activity: Napoleon Triangles, altitude 247–248 Activity: Prove It!, 657–658 Activity: The Right Triangle Fractal, 481 Activity: The Sierpin´ski Triangle, 136–137 Activity: Symbolic Proofs, 613 Activity: Three Out of Four, 189–190 of a cone, 508 of a cylinder, 407 of a parallelogram, 412 of a prism, 506 of a pyramid, 506 of a triangle, 154, 177, 401, 586 See also orthocenter Altitude Concurrency Conjecture, 177 corresponding, 126, 128, 673 definitions of, 38, 49–50 duplicating, 144 exterior. See exterior angles included, 219 incoming, 41 inscribed, 307, 319–320, 325–326, 492 interior. See interior angles linear, 50, 120–122 measure of, 39–40 naming of, by vertex, 38 obtuse, 49, 101 outgoing, 41 picture angle, 323 proving conjectures about, 680–681, 686–687 right, 49 sides of, 38 supplementary, 50 symbols for, 38 tessellations and total measure of, 379–380 transversals forming, 126–129 vertical, 50, 121, 679–680 See also polygon(s); specific polygons listed by name |
Angle Addition Postulate, 672 Angle-Angle-Angle (AAA) case, 219, 572, 574 angle bisector(s) of an angle, 40, 101, 157–158, 587–588 incenter, 177–179 of a triangle, 176–179, 586, 587–588 of a vertex angle, 242–243 Angle Bisector Concurrency Conjecture, 176 Angle Bisector Conjecture, 157 Angle Bisector/Opposite Side Activity: Toothpick Polyhedrons, analytic geometry. See coordinate Conjecture, 588 512–513 Activity: Traveling Networks, 118–119 Activity: Turning Wheels, 346–347 Activity: The Unit Circle, 651–654 Activity: Using a Clinometer, 632–633 geometry Andrade, Edna, 24 angle(s) acute, 49 addition of, 672 and arcs, 308, 319–320 base. See base angles bearing, 312 bisectors of, 40, 101, 157–158, Activity: Where the Chips Fall, 587–588 442–444 acute angle, 49 acute triangle, 60, 636, 641–642 addition of angles, 672 addition, properties of, 670 central. See central angle complementary, 50 congruent, 50 consecutive. See consecutive angles Angle Bisector Postulate, 672 Angle Bisector Theorem, 686–687 Angle Duplication Postulate, 672 angle of depression, 627 angle of elevation, 627 angle of rotation, 359 Angle-Side-Angle (ASA), 219, 225–226, 227, 574 Angles Inscribed in a Semicircle Conjecture, 320 Angola, 18, 124 angular velocity, 344 annulus, 437 Another World (Other World) (Escher), 667 I n d e x INDEX 747 antecedent, 655–656 antiprism(s), 510 Apache, 3 aperture, 264 apothem, 426 applications agriculture, 70, 435, 452, 453, 548, 648, 649 aircraft travel, 280–281, 337, 338, 340, 392, 423, 570, 644, 648, 649, 660 archaeology, 258, 276, 301, 311, 493, 638, 644, 664 architecture, 6, 9, 12, 15, 23, 25, 55, 65, 70, 83, 134, 159, 174, |
198, 204, 271, 278, 319, 345, 379, 386, 405, 414, 428, 445, 448, 449, 505, 520, 539, 548, 564, 610, 631, 660 astronomy, 40, 101, 341, 618, 620 biology, 15, 18, 268, 326, 435, 490, 597, 599–601 botany, 334, 338 business, 212, 570, 595, 609 chemistry, 9, 95, 117, 124, 187, 246, 504, 520, 536, 537, 545, 617 computers, 235 construction and maintenance, 76, 123, 134, 222, 245, 259, 278, 291, 298, 301, 318, 335, 407, 410, 414, 420, 422, 423, 424, 425, 440, 451, 457, 459, 470, 483, 495, 519, 526, 548, 556, 615, 645, 660, 685 consumer awareness, 286, 338, 458, 459, 482, 497, 518, 532, 533, 544, 555, 596, 626, 722 cooking, 458, 498, 556, 609 design, 24, 63, 70, 116, 131, 152, 175, 179, 180, 187, 193, 302, 339, 344, 351, 355, 383, 386, 387, 388, 389–391, 406, 419, 423, 424, 428, 458, 465, 483, 494, 525, 526, 533, 538, 544, 545, 549, 562, 569, 570, 608, 648, 650 distance calculations, 77, 91, 104, 150, 163, 223, 233, 240, 246, 276, 277, 318, 344, 351, 371, 391, 397, 482, 497, 498, 525, 562, 582–583, 584, 614, 621, 627, 628, 629, 638–639, 644, 645, 648, 649, 660, 677, 689 earth science, 549 engineering, 283, 291, 493, 607 environment, 519, 526 flags, 4, 26 forestry, 633 geography, 351, 442, 525, 548 government, |
675 landscape architecture, 424 law and law enforcement, 83, 100, 414 manufacturing, 34, 220, 458 maps and mapping, 36, 311, 567 medicine and health, 39, 110 meteorology and climatology, 101, 334, 483, 628 moviemaking, 563, 601 music, 339 navigation, 217, 351, 628, 629, 630, 645, 648, 660 oceanography, 576, 653 optics, 45, 52, 271, 326 photography, 263, 323, 583 physics, 45, 46, 52, 185, 271, 314, 483–484, 520, 535–537, 556, 601–602 public transit, 87 resources, consumption of, 423, 596, 649 seismology, 229 sewing and weaving, 9, 10, 14, 49, 51, 56, 60, 289, 298, 299, 415 shipping, 400, 420, 644 sports, 185, 186, 286, 315, 345, 363, 367–368, 369–370, 371, 383, 406, 435, 443, 498, 501, 562 surveying land, 36, 94, 453, 469, 649 technology, 52, 112, 131, 235, 263, 271, 283, 314, 317, 323, 339, 351, 423, 435, 498, 583, 607, 628, 630 telecommunications, 112, 435, 498 velocity and speed calculations, 134, 293, 302, 337, 338, 340, 344, 345, 351, 392, 483, 497, 660, 661 zoology and animal care, 15, 435, 536, 576, 694 approximately equal to, symbol for, 40 Arbus, Diane, 373 arc(s) addition of, 703 and angles, 308, 319–320 and chords, 308 defined, 68 endpoints of, 68 inscribed angles intercepting, 320 length of, and circumference, 341–343 major, 68, 308 measure of, 68, 319–321, 341–343 minor, 68 nautical mile and, 351 parallel lines intercepting, 321 semicircles, 68, 320, 492 symbol, 68 Arc Addition Postulate, 703 Arc Length Conjecture, 342 archaeology, 258, 276, 301, 311, 493, 638, 644, 664 arches, 271, 278, 319 Archimedean screw, |
648 Archimedean tilings, 381 Archimedes, 381, 535, 536, 616 architecture, 6, 9, 12, 15, 23, 25, 55, 65, 70, 83, 134, 159, 174, 198, 204, 271, 278, 319, 345, 379, 386, 405, 414, 428, 445, 448, 449, 505, 520, 539, 548, 564, 610, 631, 660 Archuleta-Sagel, Teresa, 265 area ancient Egyptian formula for, 453 ancient Greek formula for, 454 of an annulus, 437 of a circle, 433–434, 437–438, 492 defined, 410 of irregularly shaped figures, 411, 422, 430–432 of a kite, 418 maximizing, 421 of a parallelogram, 412–413 Pick’s formula for, 430–432 of a polygon, 411 probability and, 442–444 proportion and, 592–593 of a rectangle, 411 of a regular polygon, 426–427 of a sector of a circle, 437–438 of a segment of a circle, 437–438 surface. See surface area of a trapezoid, 417–418 of a triangle, 411, 417, 454, 634–635 units of measure of, 413 Arendt, Hannah, 634 Around the World in Eighty Days (Verne), 337 art circle designs, 10–11 Dada Movement, 260 geometric patterns in, 2–4, 9, 19 Islamic tile designs, 20–22 knot designs, 2, 16–17, 21, 396 line designs, 7–8 mandalas, 25, 576 murals, 571 op art, 3, 13–14, 66 perspective, 172 proportion and, 577, 592, 597 symmetry in, 3–4, 5 See also drawing ASA (Angle-Side-Angle), 219, 225–226, 227, 574 ASA Congruence Conjecture, 225 ASA Congruence Postulate, 673 Asian art/architecture. See China; India; Japan x e d n I 748 INDEX Assessing What You’ve Learned biology, 15, 18, 268, 326, 435, 490, Giving a Presentation, 304, 356, 408, 460, 502, 558, 618, 666, 723 Keeping a Port |
folio, 26, 92, 140, 196, 254, 304, 356, 408, 460, 502, 558, 618, 666, 723 Organize Your Notebook, 82, 140, 196, 254, 304, 356, 408, 460, 502, 558, 618, 666, 723 Performance Assessment, 196, 254, 304, 356, 460, 480, 558, 723 Write in Your Journal, 140, 196, 254, 304, 356, 408, 460, 502, 558, 666, 723 Write Test Items, 254, 356, 460, 502, 558 associative property of addition, 670 of multiplication, 670 astronomy, 40, 101, 341, 618, 620 Australia, 388 Austria, 65 auxiliary line, 200 Axis Dance Company, 660 axis of cylinder, 507 Aztecs, 25 B Babylonia, 94, 463, 469 Bacall, Lauren, 266 Bakery Counter (Thiebaud), 514 Bankhead, Tallulah, 426 base(s) of a cone, 508 of a cylinder, 507 of an isosceles triangle, 62 of a prism, 445, 506 of a pyramid, 445, 506 of a rectangle, 411 of solids, 445 of a trapezoid, 267 base angles of an isosceles trapezoid, 269 of an isosceles triangle, 62, 205–206 of a trapezoid, 267 Bauer, Rudolf, 177 Baum, L. Frank, 475 bearing, 312 “Behold” proof, 466 Belvedere (Escher), 619 Benson, Mabry, 299 Berra,Yogi, 73, 437 Bhaskara, 466 biconditional statements, 243 bilateral symmetry, 3 billiards, geometry of, 41–42, 45 binoculars, 271 597, 599–601 bisector of an angle, 40, 101, 157–158, 587–588 of a kite, 267 perpendicular. See perpendicular bisectors of a segment, 31, 147–149 of a triangle, 176–179, 586, 587–588 of a vertex angle, 242–243 Blake, William, 679 body temperature, 599–600 Bolyai, János, 718 Bookplate for Albert Ernst Bosman (Escher), 73 Borromean Rings, 18 Borromini, Francesco, 174 |
botany, 334, 338 Botswana, 364 Boulding, Kenneth, 54 Boy With Birds (Driskell), 303 Braun, Wernher von, 620 Breezing Up (Homer), 629 Brewster, David, 378 Brickwork, Alhambra (Escher), 389 Bruner, Jerome, 384 Buck, Pearl S., 398 buoyancy, 537 business, 212, 570, 595, 609 C CA Conjecture (Corresponding Angles Conjecture), 127 CA Postulate (Corresponding Angles Postulate), 673 carbon molecules, 504 Carroll, Lewis, 47, 366, 612 Carson, Rachel, 522 Cartesian coodinate geometry. See coordinate geometry ceiling of cloud formation, 628 Celsius conversion, 210 Celtic knot designs, 2, 16 center of a circle, 67, 309, 325–326 of a sphere, 507 center of gravity, 46, 184–185 Center of Gravity Conjecture, 185 center of rotation, 359 central angle of a circle, 68, 307–308 of a regular polygon, 272 centroid, 183–185, 189–190 finding, 402 Centroid Conjecture, 184 Cézanne, Paul, 504 Chagall, Marc, 55 chemistry, 9, 95, 117, 124, 187, 246, 504, 520, 536, 537, 545, 617 Chichén Itzá, 639 China, 10, 30, 36, 316, 319, 333, 463, 479, 484, 502, 520, 567 Chokwe, 18 Cholula pyramid, 527 chord(s) arcs and, 308 center of circle and, 309 central angles and, 307–308 defined, 69 perpendicular bisectors of, 309–310 properties of, 307–310 Chord Arc Conjecture, 308 Chord Central Angles Conjecture, 308 Chord Distance to a Center Conjecture, 309 Christian art and architecture, 12, 159, 174, 198, 345, 405 circle(s) annulus of concentric, 437 arcs of. See arc(s) area of, 433–434, 437–438, 492 center of, 309, 325–326 central angles of, 68, 307–308 chords of. See chord(s) circumference of, 331–333 circumscribed, 178, 179 concentric, 68 congruent, 68 cycloid, 347–348 defined, 67 diameter of |
. See diameter epicycloid, 348 equations of, 488–489 externally tangent, 315 inscribed, 179 inscribed angles of, 307, 319–320, 325–326 internally tangent, 315 in nature and art, 2, 10–11 proofs involving, 699–700, 703–704 proving properties of, 325–326 and Pythagorean Theorem, 488–489, 492 radius of, 67 rectifying a, 440 secants of, 321 sectors of, 352, 437–438 segments of, 437–438 tangent, 315 tangents to. See tangent(s) trigonometry using, 651–654 Circle Area Conjecture, 434 circumcenter defined, 177 finding, 329–330 INDEX 749 I n d e x properties of, 177–178, 189–190 in right triangle, 180 Circumcenter Conjecture, 177–178 circumference, 331–333 Circumference Conjecture, 332 circumference/diameter ratio, 331–333 circumscribed circle, 178, 179 circumscribed triangle, 71, 179 Clark, Karen Kaiser, 572 Claudel, Camille, 597 clinometer, 632–633 clockwise rotation, 359 Clothespin (Oldenburg), 615 coinciding patty papers, 147 Colette, Sidonie Gabriella, 319 collinear points, 30 commutative property of addition, 670 of multiplication, 670 compass, 7, 143 sector, 608 complementary angles, 50 composition of isometries, 374–376 computer programming, 235 computers, 235 concave kite. See dart concave polygon, 54 concentric circles, 68 concept map. See tree diagram; Venn diagram conclusion, 100, 102 concurrency, point of. See point(s) of concurrency concurrent lines, 115, 176–179 conditional proof, 655–656 conditional statement(s), 551–552 converse of, 122, 611 forming, 679 inverse of, 611 and Law of the Contrapositive, 611–612 proving. See logic; proof(s) symbol for, 552 cone(s) altitude of, 508 base of, 508 defined, 508 drawing, 81, 82 frustrums of, 451 height of, 508 radius of, 508 right, 508 similar, 593 surface area of, 448–449 vertex of, 508 volume of, 522–524 congruence of angles, 50 |
ASA, 225, 227, 673 of circles, 68 CPCTC (corresponding parts of congruent triangles are congruent), 230–231 defined, 671 diagramming of, 59 of polygons, 55 as premise of geometric proofs, 671 properties of, 671 SAA, 226, 227, 686 SAS, 221, 227, 673 of segments, 31 SSS, 220, 227, 673 symbols for, and use of, 31, 40, 59 of triangle(s), 168–169, 219–222, 225–227, 230–231 conjectures data quality and quantity and, 96 defined, 94 proving. See proof(s) See also postulates of geometry; specific conjectures listed by name Connections architecture, 9, 23, 198, 204, 271 art, 180, 345, 362, 415, 563 career, 34, 220, 235, 289, 359, 407, 422, 424, 428, 483, 519, 539, 570 consumer, 4 cultural, 25, 341, 364, 386, 414 geography, 351 history, 36, 245, 331, 360, 452, 463, 466, 469, 536, 567, 577, 638, 639, 668 language, 94, 166 literature, 337, 612 mathematics, 74, 142, 333, 381, 385, 440, 673 recreation, 63, 217, 339, 484 science, 18, 40, 52, 101, 117, 185, 271, 326, 334, 484, 504, 520, 620, 628 technology, 263, 283, 314, 498, 630, 648 consecutive angles defined, 54 of a parallelogram, 280 of a trapezoid, 268 consecutive sides, 54 consecutive vertices, 54 consequent, 655–656 constant difference, 106–108 Constantine the Great, 577 construction, 142–144 of altitudes, 154 of angle bisectors, 157–158 of angles, 144 of auxiliary lines, proofs and, 200 with compass and straightedge, 143 defined, 142–143 of a dilation design, 578–580 of an equilateral triangle, 143 Euclid and, 142 of Islamic design, 156 of an isosceles triangle, 205 of a line segment, 143 of medians, 149 of midsegments, 149 of parallel lines, 161–162 with patty paper, 143, 147 of a perpendicular bis |
ector, 147–149 of perpendiculars, 152–154 of Platonic solids, 529–530 of points of concurrency, 176–179, 183–184 of a regular hexagon, 11 of a regular pentagon, 530 of a rhombus, 287–288 of a triangle, 143, 168–169, 205 See also drawing construction and maintenance applications, 76, 123, 134, 222, 245, 259, 278, 291, 298, 301, 318, 335, 407, 410, 414, 420, 422, 423, 424, 425, 440, 451, 457, 459, 470, 483, 495, 519, 526, 548, 556, 615, 645, 660, 685 construction exercises, 24, 145, 149–150, 154–155, 158–159, 162, 169–170, 180, 181, 186, 192–193, 202, 208, 228–229, 232, 245, 252, 270, 276, 278, 281–282, 291–293, 302, 311, 312, 316, 324, 340, 345, 350–351, 355, 363, 372, 386, 400, 427, 440, 451, 479, 495, 499, 534, 545, 570, 584, 590, 591, 608, 615, 640, 645, 705 consumer awareness, 286, 338, 458, 459, 482, 497, 518, 532, 533, 544, 555, 596, 626, 722 contrapositive, 611 Contrapositive, Law of the, 612–613 converse of a statement, 122, 611 Converse of Parallel/Proportionality Conjecture, 609 Converse of the Equilateral Triangle Conjecture, 243 Converse of the Isosceles Triangle Conjecture, 206 Converse of the Parallel Lines Conjecture, 128–129 Converse of the Perpendicular Bisector Conjecture, 149 Converse of the Pythagorean Theorem, 468–470 convex polygons, 54 cooking, 458, 498, 556, 609 coordinate geometry as analytic geometry, 166 centroid, finding, 402 circumcenter, finding, 329–330 x e d n I 750 INDEX dilation in, 566–567, 617 distance in, 486–489 linear equations and, 210–211 ordered pair |
rules, 366 orthocenter, finding, 401 proofs with, 712–715 reflection in, 374–375, 467 slope and, 165–166 systems of, 401–402 translation in, 359, 366, 373 trigonometry and, 651–654 Coordinate Midpoint Property, 36, 712 Coordinate Transformations Conjecture, 367 coplanar points, 30 corresponding angles, 126, 128, 673 Corresponding Angles Conjecture (CA Conjecture), 127 Corresponding Angles Postulate (CA Postulate), 673 corresponding parts of congruent triangles are congruent (CPCTC), 230–231 cosine (cos), 621–622 Cosines, Law of, 641–643, 647 counterclockwise rotation, 359 counterexamples, 47–48 CPCTC (corresponding parts of congruent triangles are congruent), 230–231 Crawford, Ralston, 154 Crazy Horse Memorial, 569 cubic units, 514 Curie, Marie, 176 Curl-Up (Escher), 305 cyclic quadrilateral, 321 Cyclic Quadrilateral Conjecture, 321 cycloid, 347–348 cylinder(s) altitude of, 507 axis of, 507 bases of, 507 defined, 507 drawing, 81 height of, 507 oblique, 507, 516–517 radius of, 507 right, 507, 515–517 surface area of, 446–447, 459 volume of, 515–517 Czech Republic, 316 D Dada Movement, 260 daisy designs, 11 Dali, Salvador, 6 dart defined, 294 nonperiodic tessellations with, 388 properties of, 294–295 data quantity and quality, conjecture distance calculations, 77, 91, 104, and, 96 Day and Night (Escher), 407 decagon, 54 Declaration of Independence, 675 deductive reasoning, 100–102 defined, 100 geometric. See proof(s) inductive reasoning compared with, 101–102 logic as. See logic deductive system, 668 definitions, 30 imprecise, for basic concepts, 30 writing, tips on, 47–51 degree measure, 39–40 degrees, 39 dendroclimatology, 334 density, 535–536 design, 24, 63, 70, 116, 131, 152, 175, 179, 180, 187, 193, 302, 339, |
344, 351, 355, 383, 386, 387, 388, 389–391, 406, 419, 423, 424, 428, 458, 465, 483, 494, 525, 526, 533, 538, 544, 545, 549, 562, 569, 570, 608, 648, 650 deVilliers, Michael, 696 diagonal(s) of a kite, 267 of a parallelogram, 280 of a polygon, 54 of a rectangle, 289 of a rhombus, 288 of a square, 290 of a trapezoid, 269, 699 diagrams assumptions possible and not possible with, 59–60 geometric proofs with, 679 problem solving with, 73–75, 482 tree diagrams, 78 vector diagrams, 280–281 Venn diagrams, 78 See also drawing diameter defined, 67, 69 ratio to circumference, 331–333 as term, use of, 67 dice, probability and, 86 dilation(s), 566–567 constructing design with, 578–580 Dilation Similarity Conjecture, 567 direct proof, 655 disabilities, persons with, 483, 660 displacement, 535–536 dissection, 462 distance in coordinate geometry, 486–489 defined, from point to line, 154 of a translation, 358 150, 163, 223, 233, 240, 246, 276, 277, 318, 344, 351, 371, 391, 397, 482, 497, 498, 525, 562, 582–583, 584, 614, 621, 627, 628, 629, 638–639, 644, 645, 648, 649, 660, 677, 689 distance formula, 486–488, 712 distributive property, 670 division property of equality, 670 dodecagon, 54, 380 dodecahedron, 505 Dodgson, Charles Lutwidge, 612 See also Carroll, Lewis Doren, Albert Van, 198 Double-edged Straightedge Conjecture, 287 double negation, 552 doubling the angle on the bow, 217 Doyle, Sir Arthur Conan, 698 drawing circle designs, 10–11 congruence markings in, 31, 59 daisy designs, 11 defined, 142 equilateral triangle, 142 geometric solids, 80–82 isometric, 80–82, 539–541 knot designs, 16 line designs, |
8 mandalas, 25 op art, 13–14, 66 orthographic, 539–541 perspective, 172–175 polygons, regular, 272 to scale, 566–567 tessellations, 21, 389–391, 393–396 See also construction Drawing Hands (Escher), 141 Driskell, David C., 303 dual of a tessellation, 382 Dudeney, Henry E., 490 Dukes, Pam, 315 duplication of geometric figures, 142–144 E earth science, 549 Ebner-Eschenbach, Marie von, 80 edge, 505 Edison, Thomas, 339 Egypt, 2, 36, 94, 245, 333, 386, 453, 463, 469, 519, 583, 648 Einstein, Albert, 393 Elements (Euclid), 142, 668, 671 elimination method of solving equation systems, 285–286, 401–402 elliptic geometry, 719–720 INDEX 751 Emerging Order (Hoch), 260 endpoints of an arc, 68 of a line segment, 31 engineering, 283, 291, 493, 607 England, 38, 703 environment, 519, 526 epicycloid, 348 equal symbol, use of, 31 equal to, symbol for, 40 equality, properties of, 670–671 equations of a circle, 488–489 linear. See linear equations equator, 351, 507 equiangular polygon, 55, 261 Equiangular Polygon Conjecture, 261 equiangular triangle. See equilateral triangle(s) Equiangular Triangle Conjecture, 243 equidistant, defined, 148 Equilateral/Equiangular Triangle Conjecture, 242–243 equilateral polygon, 55 equilateral triangle(s) angle measures of, 242 constructing, 143 defined, 61 drawing, 142 as equiangular, 242–243 as isosceles triangle, 205 Napoleon’s theorem and, 247–248 properties of, 205–206 tessellations with, 379–381, 394–395 and 30°-60°-90° triangles, 476 Eratosthenes, 344 Escher, M. C., 1, 27, 93, 141, 197, 255, 305, 357, 409, 462, 503, 559, 619, 667 dilation and rotation designs of, 578 knot designs of, 17 and Napoleon’s The |
orem, 248 and tessellations, 389, 393–395, 398–399, 407 Euclid, 142, 463, 668, 671, 673, 718 Euclidean geometry, 142, 668, 668–674, 718, 720 Euler, Leonhard, 118, 189, 336, 512 Euler line, 189–190 Euler Line Conjecture, 189 Euler segment, 189–190 Euler Segment Conjecture, 190 Euler’s Formula for Networks, 118–119 Euler’s Formula for Polyhedrons, 512–513 Euler’s magic square, 336 European art and architecture, 2, 5, 25, 38, 55, 62, 65, 159, 174, 278, 316, 405, 440, 703 Evans, Minnie, 404 Explorations Alternative Area Formulas, 453–454 Constructing a Dilation Design, 578 Cycloids, 346–348 The Euler Line, 189–190 Euler’s Formula for Polyhedrons, 512–513 The Five Platonic Solids, 528–530 Geometric Probability I, 86–87 Geometric Probability II, 442–444 Indirect Measurement, 632–633 Ladder Climb, 491 Napoleon’s Theorem, 247–248 Non-Euclidean Geometries, 718–720 Orthographic Drawing, 539–541 Patterns in Fractals, 135–137 Perspective Drawing, 172–175 Pick’s Formula for Area, 430–432 Proof as Challenge and Discovery, 696–697 A Pythagorean Fractal, 480–481 The Seven Bridges of Königsberg, 118–119 Sherlock Holmes and Forms of Valid Reasoning, 551–553 Star Polygons, 264–265 Three Types of Proofs, 655–658 Trigonometric Ratios and the Unit Circle, 651–654 Two More Forms of Valid Reasoning, 611–613 Why Elephants Have Big Ears, 599–602 Extended Parallel/Proportionality Conjecture, 606, 609 Exterior Angle Sum Conjecture, 261 exterior angles alternate, 126–129 of a polygon, 260–261 sum of, 261 transversal forming, 126 of a triangle, 215–216 externally tangent circles, 315 Exxon Valdez, 519 F face(s) and area, 445 lateral, |
uclidean, 718–720 as word, 94 Gerdes, Paulus, 124 Germain, Sophie, 74 Giovanni, Nikki, 325 Girodet, Anne-Louis, 247 given. See antecedent glide reflection, 376, 398–399 glide-reflectional symmetry, 376 Goethe, Johann Wolfgang von, 486 golden cut, 585 golden ratio, 585, 598, 610 golden rectangle, 610 golden spiral, 610 Goldman, Emma, 686 Goldsworthy, Andy, 5 Golomb, Solomon, 53 Gordian knot, 18 gou gu, 479, 502 Graphing Calculator projects Drawing Regular Polygons, 272 Line Designs, 132 Lines and Isosceles Triangles, 248 Maximizing Area, 421 Maximizing Volume, 538 Trigonometric Functions, 654 graphs and graphing intersections of lines found by, 211 of linear equations, 210–211 of linear functions, 108 of system of equations, 285 of trigonometric functions, 654 See also Graphing Calculator projects gravity and air resistance, 601–602 gravity, center of, 46, 184–185 great circle(s), 351, 507, 719–720 Great Pyramid of Khufu, 519, 527, 683 Greeks, ancient, 18, 30, 36, 142, 233, 344, 381, 425, 454, 463, 528, 610, 616, 620, 668 Greve, Gerrit, 306 Guatemala, 7, 670 H Haldane, J. B. S., 592 Hand with Reflecting Sphere (Escher), 93 Hansberry, Lorraine, 581 Harlequin (Vasarely), 13 Harryhausen, Ray, 601 Hawaii, 452 height of a cone, 508 of a cylinder, 507 of a parallelogram, 412 of a prism, 506 of a pyramid, 447, 506 of a rectangle, 411 of a trapezoid, 417 of a triangle, 154, 634–635 Hein, Piet, 2, 183, 511, 563 hemisphere(s) defined, 507 drawing, 81, 82 volume of, 542–543 heptagon, 54, 426 heptahedron, 505 Hero, 425, 454 Hero’s formula for area, 454 Hesitate (Riley), 13 hexagon(s), 54 in nature and art, 2, 21, 379 regular. See regular hexagon |
EX 753 Cover the Square, 454 Dissecting a Hexagon I, 203 Dissecting a Hexagon II, 263 Equal Distances, 85 Folding Cubes I, 151 Folding Cubes II, 474 Fold, Punch, and Snip, 485 Four-Way Split, 425 Hexominoes, 79 Mental Blocks, 695 Moving Coins, 452 Mudville Monsters, 479 Net Puzzle, 259 Painted Faces I, 403 Painted Faces II, 602 Patchwork Cubes, 545 Pentominoes I, 58 Pentominoes II, 117 Pickup Sticks, 6 Picture Patterns I, 340 Picture Patterns II, 387 Picture Patterns III, 691 Piet Hein’s Puzzle, 511 Polyominoes, 53 The Puzzle Lock, 182 A Puzzle Quilt, 284 Puzzle Shapes, 633 Random Points, 436 Rolling Quarters, 328 Rope Tricks, 640 Rotating Gears, 105 The Spider and the Fly, 490 The Squared Square Puzzle, 429 3-by-3 Inductive Reasoning Puzzle I, 392 3-by-3 Inductive Reasoning Puzzle II, 626 TIC-TAC-NO!, 585 Visual Analogies, 164 incenter, 177–179 Incenter Conjecture, 178–179 inclined plane, 483–484, 660 included angle, 219 included side, 219 incoming angle, 41 India, 6, 25, 333, 466, 616 indirect measurement with similar triangles, 581–582 with string and ruler, 584 with trigonometry, 621 indirect proof, 656, 698–700 inductive reasoning, 94–96 deductive reasoning compared to, 101–102 defined, 94 figurate numbers, 115 finding nth term, 106–108 mathematical modeling, 112–115 inequalities, triangle, 213–216 inscribed angle(s), 307, 319–320, 325–326, 492 Inscribed Angle Conjecture, 319, 325–326 Inscribed Angle Theorem, 703 Inscribed Angles Intercepting Arcs Conjecture, 320 inscribed circle, 179 inscribed quadrilateral, 321 inscribed triangle, 71, 179 integers Pythagorean triples, 468–469 rule generating, 106–108 See also numbers intercepted arc, 320, 321 interior angles adjacent, 215–216 alternate, 126–129, 681 of polygons, 256–257, 261 remote, 215–216 transversal forming, 126 interior design, 428 |
internally tangent circles, 315 intersection of lines finding, 211 reflection images and, 375 See also perpendicular lines; point(s) of concurrency Interwoven Patterns–V (Roelofs), 396 inverse cosine (cos1), 624 inverse of a conditional statement, 611 inverse sine (sin1), 624 inverse tangent (tan1), 624 Investigations Angle Bisecting by Folding, 157 Angle Bisecting with Compass, 158 Chords and Their Central Angles, 307–308 Concurrence, 176–177 Constructing Parallel Lines by Folding, 161 Copying a Segment, 143–144 Copying an Angle, 144 Corresponding Parts, 586 Cyclic Quadrilaterals, 321 Defining Angles, 49–50 Defining Circle Terms, 69 Dilations on a Coordinate Plane, 566–567 The Distance Formula, 486–487 Do All Quadrilaterals Tessellate?, 385 Do All Triangles Tessellate?, 384 Do Rectangle Diagonals Have Special Properties?, 289 Do Rhombus Diagonals Have Special Properties?, 288 The Equation of a Circle, 488 Extended Parallel/ Proportionality, 606 Exterior Angles of a Triangle, 215–216 Finding a Minimal Path, 367–368 Finding the Arcs, 342 Finding the Right Bisector, 147–148 Finding the Right Line, 152–153 Finding the Rule, 106–107 The Formula for the Surface Area of a Sphere, 546–547 The Formula for the Volume of a Sphere, 542 Angles Inscribed in a Semicircle, Four Parallelogram Properties, 320 Arcs by Parallel Lines, 321 Are Medians Concurrent?, 183–184 Area Formula for Circles, 433–434 Area Formula for Kites, 418 Area Formula for Parallelograms, 412 Area Formula for Regular Polygons, 426–427 Area Formula for Trapezoids, 417–418 Area Formula for Triangles, 417 Area of a Triangle, 634–635 Area Ratios, 592–593 Balancing Act, 184–185 Base Angles in an Isosceles Triangle, 205 The Basic Property of a Reflection, 360–361 Can You Prove the SSS Similarity Conjecture?, 708 279–280 Going Off on a Tangent, 313 How Do We Define Angles in a Circle?, 307 Incenter and Circumcenter, 177–178 Inscribed Angle Properties, 319 |
Inscribed Angles Intercepting the Same Arc, 320 Is AA a Similarity Shortcut?, 572 Is ASA a Congruence Shortcut?, 225 Is SAS a Congruence Shortcut?, 221 Is SAS a Similarity Shortcut?, 574 Is SSS a Congruence Shortcut?, 220 Is SSS a Similarity Shortcut?, 573 Is the Converse True?, 128–129, 206, 468–469 Is There a Polygon Sum Formula?, 256–257 Chords and the Center of the Is There an Exterior Angle Sum?, Circle, 309 260–261 x e d n I 754 INDEX Isosceles Right Triangles, 475 The Law of Sines, 635 The Linear Pair Conjecture, 120 Mathematical Models, 29 Mirror, Mirror, 581 Opposite Side Ratios, 588 Overlapping Segments, 102 Parallels and Proportionality, 604–605 Party Handshakes, 112–114 Patty-Paper Perpendiculars, 153 Perpendicular Bisector of a Chord, 309–310 Proving Parallelogram Conjectures, 692–693 Proving the Tangent Conjecture, 699–700 A Pythagorean Identity, 641 Reflections over Two Intersecting Lines, 375 Reflections over Two Parallel Lines, 374 Right Down the Middle, 148–149 The Semiregular Tessellations, 380–381 Shape Shifters, 96 Solving Problems with Area Formulas, 422 Space Geometry, 82 Surface Area of a Cone, 449 Surface Area of a Regular Pyramid, 448 The Symmetry Line in an Isosceles Triangle, 242 Tangent Segments, 314 A Taste of Pi, 332 30°-60°-90° Triangles, 476 The Three Sides of a Right Triangle, 462–463 Transformations of a Coordinate Plane, 367 Trapezoid Midsegment Properties, 274–275 Triangle Midsegment Properties, 273–274 The Triangle Sum, 199–200 Triangles and Special Quadrilaterals, 60–61 Trigonometric Tables, 622–623 Vertical Angles Conjecture, 121 Virtual Pool, 41–42 The Volume Formula for Prisms and Cylinders, 515–516 The Volume Formula for Pyramids and Cones, 522 Volume Ratios, 594 What Are Some Properties of Kites?, 266–267 What Are Some Properties of Trapezoids?, 268– |
269 What Can You Draw with the Double-Edged Straightedge?, 287 What Is the Shortest Path from A to B?, 214 What Makes Polygons Similar?, 564 Where Are the Largest and Smallest Angles?, 215 Which Angles Are Congruent?, 126–128 Iran, 358, 448 irrational numbers, 333 Islamic art and architecture, 2, 6, 10, 20–23, 60, 156, 207, 298, 379, 389, 448, 520 isometric drawing, 80–82, 539–541 isometry composition of, 373–376 on coordinate plane, 359, 366–367, 373, 467 defined, 358 properties of, 366–370 types of. See reflection; rotation; translation isosceles right triangle, 475–476 Isosceles Right Triangle Conjecture, 475 isosceles trapezoid(s), 268–269 Isosceles Trapezoid Conjecture, 269 Isosceles Trapezoid Diagonals Conjecture, 269 isosceles triangle(s), 204–206 base angles of, 62, 205–206 base of, 62 construction of, 205 defined, 61 legs of, 204 proofs involving, 698–699 right, 475–476 vertex angle bisector, 242–243 vertex angle of, 62 See also equilateral triangle(s) Isosceles Triangle Conjecture, 205 converse of, 206 Israel, 20, 505 Italy, 18, 159, 174, 463, 525, 549, 639, 668 J Jainism, 616 Jamaica, 4 James Fort, 638 James III, Richard, 385 Japan, 4, 7, 14, 17, 19, 83, 299, 386, 387, 414, 525, 646, 720 Jefferson, Thomas, 256 Jewish art, 405 Jimenez, Soraya, 435 Jiuzhang suanshu, 479 John Paul I (pope), 260 journal, defined, 140 Joyce, Bruce, 94 Joyce, James, 603 Juchi, Hajime, 14 K kaleidoscopes, 378 Kamehameha I, 452 Keller, Helen, 445 Kickapoo, 548 Kim, Scott, 383 King (Evans), 404 King Kong, 601 King, Martin Luther, Jr., 152 kite(s) area of, 418 concave. See dart defined, 63 nonperiodic tessellations with, 388 nonvertex angles of, 266 properties |
of, 266–267 tessellations with, 398 vertex angles of, 266 Kite Angle Bisector Conjecture, 267 Kite Angles Conjecture, 267 Kite Area Conjecture, 418 Kite Diagonal Bisector Conjecture, 267 Kite Diagonals Conjecture, 267 kites (recreational), 63, 419 Klee, Paul, 433 knot designs, 2, 16–17, 21, 396 koban (Japanese architecture), 83, 414 Königsberg, 118 L La Petite Chatelaine (Claudel), 597 Lahori, Ustad Ahmad, 6 landscape architecture, 424 Laos, 363 lateral edges oblique, and volume, 516 of a prism, 506 of a pyramid, 506 lateral faces of a prism, 445, 506 of a pyramid, 445 and surface area, 445 latitude, 507 Lauretta, Sister Mary, 147 law, 100 law enforcement, 83, 414 Law of Cosines, 641–643, 647 Law of Sines, 634–637, 647 I n d e x INDEX 755 x e d n I Law of Syllogism, 611, 612–613 Law of the Contrapositive, 612–613 Lec, Stanislaw J., 13 legs of an isosceles triangle, 204 of a right triangle, 462 lemma, 692 L’Engle, Madeleine, 100 length of an arc, 341–343 symbols to indicate, 31 Leonardo da Vinci, 331, 360, 440, 463 LeWitt, Sol, 61 light, angles of, 45, 52 Lin, Maya, 130 line(s) auxiliary, 200 concurrent, 115, 176–179 description of, 28, 29–30 diagrams of, 59 equations of. See linear equations intersecting. See intersection of lines models of, 28–29 naming of, 28 in non-Euclidean geometries, 718–720 parallel. See parallel lines perpendicular. See perpendicular lines postulates of, 672 segments of. See line segment(s) skew, 48 slope of. See slope symbol for, 28 tangent. See tangent(s) transversal, 126 line designs, 7–8 Line Intersection Postulate, 672 line of best fit, 111 line of reflection, 360–361 line of symmetry, 3, 361 Line Postulate, 672 line segment(s) |
, 648, 660 Nazca Lines, 567 negation, 552 nets, 78, 528–530 networks, 118–119 New Zealand, 709 n-gon, 54, 257 See also polygon(s) Nigeria, 16 nonagon, 54 non-Euclidean geometries, 718–720 nonperiodic tiling, 388 nonrigid transformation, 358, 566–567, 578–580 nonvertex angles, 266 notebook, defined, 92 nth term, finding, 106–108 numbers integers. See integers irrational, 333 rectangular, 115 rounding, 423 square roots, 473–474 triangular, 115 numerical name for tiling, 380 O Oates, Joyce Carol, 10 oblique cylinder, 507, 516–517 oblique prism, 506, 516, 517 oblique triangular prism, 506 obtuse angle, 49, 101 obtuse triangle, 60, 641–642 oceanography, 576, 653 octagon, 54, 380 octahedron, 528 odd numbers O’Hare, Nesli, 41 oil spills, 519, 526 Oldenburg, Claes, 615 100 Cans (Warhol), 507 one-point perspective, 173 op art, 3, 13–14, 66 opposite angle, 215, 620 opposite angles of a parallelogram, 279, 692–693 opposite side, trigonometry, 620 opposite sides of a parallelogram, 280, 693–694 optics, 45, 52, 271, 326 ordered-pair rules, 366 orthocenter defined, 177 P Palestinian art, 10 paragraph proofs, 122 circle conjectures, procedures for, 326 quadrilateral conjectures, procedures for, 295 parallel lines arcs on circle and, 321 construction of, 161–162 defined, 48, 161 diagrams of, 59 proofs involving, 680–681 properties of, 127–129, 672 proportional segments by, 603–607 and reflection images, 374 slope of, 165–166, 712 symbol for, 48 Parallel Lines Conjecture, 127–128 converse of, 128–129 Parallel Lines Intercepted Arcs Conjecture, 321 Parallel/Proportionality Conjecture, 605 converse of, 606, 617 Extended Parallel/Proportionality, 606, 609 Parallel Slope Property, 165, 712 parallelogram(s) altitude of, 412 area of, |
412–413 defined, 63 height of, 412 proofs involving, 692–693, 696–697 properties of, 279–280, 287–290 relationships of, 78 special, 287–290 tessellations with, 399 in vector diagrams, 280–281 See also specific parallelograms listed by name Parallelogram Area Conjecture, 412 Parallelogram Consecutive Angles Conjecture, 280 Parallelogram Diagonal Lemma, 692 Parallelogram Diagonals Conjecture, 280 Parallelogram Opposite Angles Conjecture, 279, 692–693 Parallelogram Opposite Sides Conjecture, 280, 693–694 Parallel Postulate, 672, 718–719 partial mirror, 52 Path of Life I (Escher), 559, 578 patterns. See inductive reasoning patty papers, 142, 143, 147 Pei, I. M., 204, 445 Pelli, Cesar, 23 Penrose, Sir Roger, 388 Penrose tilings, 388 pentagon(s) area of, 426 congruent, 55 naming of, 54 in nature and art, 2 sum of angles of, 256–257 tessellations with, 379, 385, 386 Pentagon Sum Conjecture, 256 performance assessment, 196 period, 654 periodic curve, 348 periodic phenomena, trigonometry and, 654 periscope, 131 perpendicular bisector(s) of a chord, 309–310 constructing, 147–149 defined, 147 linear equations for, 211 of a rhombus, 288 of a triangle, 149, 176–178 See also bisector; circumcenter; perpendiculars Perpendicular Bisector Concurrency Conjecture, 177 Perpendicular Bisector Conjecture, 148 Converse of, 149 Perpendicular Bisector of a Chord Conjecture, 310 perpendicular lines defined, 48 diagrams of, 59 slope of, 165–166, 712 symbol for, 48 Perpendicular Postulate, 672 Perpendicular Slope Property, 165, 712 Perpendicular to a Chord Conjecture, 309 perpendiculars constructing, 152–154 defined, 152 postulate of, 672 perspective, 172–175 Peru, 567 Phelps, Elizabeth Stuart, 204 photography, 263, 323, 583 physics, 45, 46, 52, 185, 271, 314, 483–484, 520, 535–537, 556 |
, 601–602 pi, 331–333, 337 symbol for, 331 INDEX 757 I n d e x Picasso, Pablo, 341 Pick, Georg Alexander, 430 Pickford, Mary, 531 Pick’s formula, 430–432 picture angle, 323 pinhole camera, 583 plane(s) concurrent, 176 naming of, 28 as undefined term, 28–30 plane figures. See geometric figures Plato, 528, 668 Platonic solids (regular polyhedrons), 505, 528–530 plumb level, 245 point(s) collinear, 30 coplanar, 30 description of, 28, 29–30 diagrams of, 59 locus of, 75 models of, 28–29 naming of, 28 symbol for, 28 point(s) of concurrency, 176–179, 183–185 centroid, 183–185, 189–190, 402 circumcenter. See circumcenter construction of, 176–179, 183–184 coordinates for, finding, 329–330, 401, 402 defined, 176 incenter, 177–179 orthocenter. See orthocenter point of tangency, 69 point symmetry, 361 pollution, 519, 526 Polya, George, 234, 668 polygon(s), 54–55 area of, 411 circumscribed circles and, 179 classification of, 54 concave, 54 congruent, 55 consecutive vertices of, 54 convex, 54 defined, 54 diagonals of, 54 drawing, 272 equiangular, 55 equilateral, 55 exterior angles of, 260–261 inscribed circles and, 179 interior angles of, 256–257, 261 naming of, 54 regular. See regular polygon(s) sides of, 54 similar. See similarity sum of angle measures, 256–257, 260–261 symmetries of, 362 tesellations with, 379–381, 384–385 vertex of a, 54 See also specific polygons listed by name Polygon Sum Conjecture, 257 polyhedron(s) defined, 505 edges of, 505 Euler’s formula for, 512–513 faces of, 505 models of, building, 512–513, 528–530 naming of, 505 nets of, 78, 528–530 regular, 505 similar, 593 vertex of a, 505 See also solid(s); specific polyhedrons listed by name Polynesian navigation, 630 pool, geometry of, 41–42, 45 portfolio, |
used in, 692 logical family tree used in, 682–684 paragraph. See paragraph proofs planning and writing of, 294–295, 679–684, 687–688 postulates of, 668, 671–673, 703, 706, 718–719 premises of, 668, 669–674, 680, 712 of the Pythagorean Theorem, 463–464, 466 of quadrilateral conjectures, 294–295, 692–693, 696–697, 699, 712–715 of similarity, 706–709 x e d n I 758 INDEX symbols and symbolic form of, 551–553, 611, 612, 613 of triangle conjectures, 681–684, 686–688, 706–709 two-column, 655, 687–688 See also logic proportion, 560–561 with area, 592–593 of corresponding parts of triangles, 586–588 defined, 560 indirect measurement and, 581–582 of segments by parallel lines, 603–607 similarity and, 560–561, 565–566, 586–588 with volume, 593–594 See also ratio; trigonometry Proportional Areas Conjecture, 593 proportional dividers, 609 Proportional Parts Conjecture, 586 Proportional Volumes Conjecture, 594 protractor angle measure with, 39, 142 and clinometers, 632–633 construction as not using, 143 defined, 47 Ptolemy, Claudius, 620 public transit, 87 Pushkin, Aleksandr, 38 puzzles. See Improving Your Algebra Skills; Improving Your Reasoning Skills; Improving Your Visual Thinking Skills pyramid(s) altitude of, 506 base of, 445, 506 defined, 506 drawing, 81 height of, 447, 506 lateral faces of, 445 slant height of, 447 surface area of, 446, 447–448 truncated, 685 vertex of, 506 volume of, 522–524 Pyramid-Cone Volume Conjecture, 522 pyramids (architectural structures), 62, 204, 245, 445, 447, 519, 527, 583, 639, 644 Pythagoras of Samos, 463, 668 Pythagorean Identity, 641 Pythagorean Proposition (Loomis |
), 463 Pythagorean Theorem, 462–464 and circle equations, 488–489 circles and, 492 converse of, 468–470 cultural awareness of principle ratio of, 463, 469 and distance formula, 486–488, 712 fractal based on, 480–481 and isosceles right triangle, 475–476 and Law of Cosines, 641–642 picture representations of, 462, 480 problem solving with, 482 proofs of, 463–464, 466 Pythagorean identities and, 641 and similarity, 480 statement of, 463 and 30°-60°-90° triangle, 476–477 trigonometry and, relationship of, 641 Pythagorean triples, 468–469 Q Q.E.D., defined, 295 quadrilateral(s) area of, ancient Egyptian formula for, 453 congruent, 55 cyclic, 321 definitions of, 62–64 linkages of, 283 naming of, 54 proofs involving, 294–295, 692–693, 696–697, 699, 712–715 proving properties of, 294–295 special, 62–64 sum of angles of, 256–257 tessellations with, 379, 385 See also specific quadrilaterals listed by name Quadrilateral Sum Conjecture, 256 quilts and quiltmaking, 9, 56, 284, 299, 415, 519 R racetrack geometry, 345 radical expressions, 473–474 radiosonde, 628 radius and arc length, 342–343 of a circle, 67 and circumference formula, 332 of a cone, 508 of a cylinder, 507 of a sphere, 507 tangents and, 313–314 as term, use of, 67 circumference/diameter, 331–333 defined, 560 equal. See proportion of Euler segment, 189–190 golden, 585, 610 probability, 86 slope. See slope trigonometric. See trigonometry See also similarity ray(s) angles defined by, 38 concurrent, 176 defined, 32 naming of, 32 symbol of, 32 reasoning deductive. See logic inductive. See inductive reasoning record players, 339 rectangle(s) area of, 411 base of, 411 defined, 63 golden, 610 height of, 411 properties of, 289 sum of angles of, 289 Rectangle Area Conjecture, 411 Rectangle Diagonals Conjecture, 289 rectangular |
numbers, 115 rectangular prism, 80, 446, 506 rectangular solid(s), drawing, 80 rectifying shapes, 440 recursive rules, 135–137 Red and Blue Puzzle (Benson), 299 reflection composition of isometries and, 374–376 defined, 360 glide, 376, 398–399 line of, 360–361 minimal path and, 367–370 as type of isometry, 358 Reflection Line Conjecture, 361 reflectional symmetry, 3, 4, 361 Reflections over Intersecting Lines Conjecture, 375 Reflections over Parallel Lines Conjecture, 374 reflexive property of congruence, 671 reflexive property of equality, 670 reflexive property of similarity, 706 regular dodecagon regular heptagon, 426 tessellations with, 380 regular dodecahedron, 505, 528–530 regular hexagon(s), 11 area of, 426 INDEX 759 tessellations with, 379–381, 390, 393–394 regular hexahedron, 528–530 regular icosahedron, 528–530 regular octagon, 380 regular octahedron, 528–530 regular pentagon, 362, 379, 426, 530 regular polygon(s) apothem of, 426 area of, 426–427 defined, 55 drawing, 272 symmetries of, 362 tessellations with, 379–381 Regular Polygon Area Conjecture, 427 regular polyhedrons (Platonic solids), 505, 528–530 regular tessellation, 380 regular tetrahedron, 528–530 remote interior angles, 215–216 Renaissance, 172 Reptiles (Escher), 394 resources, consumption of, 423, 596, 649 resultant vector, 281 revolution, solid of, 84 rhombus(es) construction of, 287–288 defined, 63 properties of, 287–288 Rhombus Angles Conjecture, 288 Rhombus Diagonals Conjecture, 288 Rice, Marjorie, 385, 386 right angle, 49 right cone, 508, 593 right cylinder, 507, 515–517 right pentagonal prism, 506 right prism, 506, 515–517, 593 right triangle(s) adjacent side of, 620 defined, 60 hypotenuse of, 462 isosceles, 475–476 legs of, 462 opposite side of, 620 properties of. See |
Pythagorean Theorem; trigonometry similarity of, 590–591 30°-60°-90° type, 476–477 rigid transformation. See isometry Riley, Bridget, 13 Roelofs, Rinus, 396 Romans, ancient, 36, 271, 278 Roosevelt, Eleanor, 492 Rosten, Leo, 287 rotation defined, 359 model of, 359 spiral similarity, 580 tessellations by, 393–395 as type of isometry, 358 rotational symmetry, 3–4, 361 rounding numbers, 423 Rounds and Triangles (Bauer), 177 ruler, 7, 142, 143, 584 Rumi, Jalaluddin, 20 Russell, Bertrand, 692 Russia, 17, 118 S SAA Congruence Conjecture, 226 SAA Congruence Theorem, 686 sailing. See navigation sangaku, 646 Sarton, May, 379 SAS Congruence Conjecture, 221 SAS Congruence Postulate, 673 SAS Similarity Conjecture, 574 SAS Similarity Theorem, 706–707 SAS Triangle Area Conjecture, 635 satellites, 314, 317 Savant, Marilyn vos, 76 scale drawings, 566–567 scale factor, 566 scalene triangle, 384 Schattschneider, Doris, 385, 580 sculpture, geometry and, 577, 592, 597, 616 secant, 321 section, 84 sector compass, 608 sector of a circle, 352 area of, 437–438 segment(s) of a circle, 437–438 in elliptic geometry, 719 line. See line segment(s) Segment Addition Postulate, 672 segment bisector, 147 Segment Duplication Postulate, 672 seismology, 229 self-similarity, 135 semicircle(s), 68, 320, 492 semiregular tessellation, 380–381 sewing and weaving, 9, 10, 14, 49, 51, 56, 60, 289, 298, 299, 415 shadows, indirect measurement and, 581–582 Shah Jahan, 6 Shintoism, 646 shipping, 400, 420, 644 Shortest Distance Conjecture, 153 show. See consequent Sicily, 668 side(s) adjacent, 620 of an angle, 38 included, 219 opposite, 620 of a parallelogram, 280, 293–294 of a polygon, 54 Side-Ang |
le-Angle (SAA), 219, 226, 227, 574 Side-Angle Inequality Conjecture, 215 Side-Angle-Side (SAS), 219, 220, 227, 574, 642 Side-Side-Angle (SSA) case, 219, 221–222, 574, 636 Side-Side-Side (SSS), 219, 220, 227, 573, 642 Sierpin´ski tetrahedron, 204 Sierpin´ski triangle, 135–137 Sills, Beverly, 482 similarity, 563–566 and area, 592–593 defined, 565 dilation and, 566–567, 578–580 indirect measurement and, 581–582 mural making and, 571 and polygons, 563–567 proofs involving, 706–709 properties of, 706 and Pythagorean fractal, 480 and Pythagorean Theorem, 480 ratio and proportion in, 560–561, 565–566, 586–588 segments and, 603–607 of solids, 593 spiral, 580 symbol for, 565 of triangles, 200–201, 572–574, 586–588 trigonometry and, 620 and volume, 593–594 simplification of radical expressions, 473 sine (sin), 621–622 Sines, Law of, 634–637, 647 Sioux, 449 sketching, 142 See also construction; drawing skew lines, 48 slant height, 447 slope calculating, 133–134 defined, 133 formula, 133 of parallel lines, 165–166, 712 of perpendicular lines, 165–166, 712 slope-intercept form, 210–211 slope-intercept form, 210–211 slope triangle, 133 smoking cigarettes, 722 Snakes (Escher), 17 solar technology, 423 solid(s) bases of, 445 with curved surfaces, 506–508 drawing, 80–82 faces of, 505 760 INDEX lateral faces of, 445, 506 nets of, 78, 528–530 Platonic, 505, 528–530 of revolution, 84 section, 84 similar, 593–594 surface area of. See surface area See also polyhedron(s); specific solids listed by name solid of revolution, 84 South America, 101 space, 80–82 defined, 80 See also solid(s) Spain, 15, |
20 speed. See velocity and speed calculations Spenger, Sylvia, 18 sphere(s) center of, 507 coordinates on, 719–720 defined, 507 drawing, 81, 82 elliptic geometry and, 719–720 great circles of, 351, 507, 719–720 hemisphere. See hemisphere radius of, 507 surface area of, 546–547 volume of, 542–543 Sphere Surface Area Conjecture, 547 Sphere Volume Conjecture, 542 spherical coordinates, 719–720 spiral(s) golden, 610 in nature and art, 35, 65 spiral similarity, 580 sports, 185, 186, 286, 315, 345, 363, 367–368, 369–370, 371, 383, 406, 435, 443, 498, 501, 562 Sproles, Judy, 279 square(s) defined, 47–48, 64 proofs involving, 712–715 properties of, 290 symmetry of, 362 tessellations with, 379–381, 389 Square Diagonals Conjecture, 290 Square Diagonals Theorem, 712–715 Square Limit (Escher), 409 square pyramid, 506 square root property of equality, 670 square roots, 473–474 square units, 413 Sri Lanka, 451 SSS Congruence Conjecture, 220 SSS Congruence Postulate, 673 SSS Similarity Conjecture, 573 SSS Similarity Theorem, 708–709 star polygons, 264–265 steering linkage, 283 Steinem, Gloria, 225 Still Life and Street (Escher), 255 Still Life With a Basket (Cézanne), 504 Stonehenge, 703 straightedge, 7, 143 double-edged, 287–288 substitution method of solving equation systems, 285–286, 402 subtraction property of equality, 670 Sumners, DeWitt, 18 Sunday Morning Mayflower Hotel (Hockney), 540 supplementary angles, 50 surface area of a cone, 448–449 of a cylinder, 446–447, 459 defined, 445 of a prism, 446, 459 of a pyramid, 445, 447–448 of a sphere, 546–547 and volume, relationship of, 599–602 surfaces, area and, 445 surveying land, 36, 94, 453, 469, 649 Swenson, Sue, 422 Syllogism, Law of, 611, 6 |
12–613 Symbolic Logic, Part I (Dodgson), 612 symbols angle, 38 approximately equal to, 40 arc, 68 conditional statement, 552 congruence, 31, 40, 59 equals, use of, 31 glide reflection, 398 image point label, 358 line, 28 line segment, 31 of logic, 551–553, 611, 612, 613 measure, 31, 40 negation (logic), 552 parallel, 48 perpendicular, 48 pi, 331 plane, 28 point, 28 ray, 32 same measure, 31 similarity, 565 slant height, 447 therefore, 552 triangle, 54 symmetric property of congruence, 671 symmetric property of equality, 670 symmetry, 3–4, 361–362 bilateral, 3 glide-reflectional, 376 in an isosceles trapezoid, 269 in an isosceles triangle, 242 in a kite, 266 line of, 3, 361 of polygons, 362 reflectional, 3, 4, 361 rotational, 3–4, 361 Symmetry Drawing E25 (Escher), 393 Symmetry Drawing E99 (Escher), 395 Symmetry Drawing E103 (Escher), 197 Symmetry Drawing E105 (Escher), 389 Symmetry Drawing E108 (Escher), 399 systems of linear equations, 285–286, 401–402 Szent-Györgyi, Albert, 120 Szyk, Arthur, 675 T Taj Mahal, 6 Take Another Look angle measures, 253 area, 459 circumference, 355 congruence shortcuts, 253 cyclic quadrilaterals, 355 dilation, 617 polygon conjectures, 303–304 Pythagorean Theorem, 501–502 quadrilateral conjectures, 253 similarity, 617–618 tangents, 355 triangle conjectures, 253, 303 trigonometry, 665–666 volume, 557–558 Talmud, 417 Tan, Amy, 67 tangent(s) defined, 69 external, 315 internal, 315 point of, 69 proofs involving, 699–700 properties of, 313–315 and Pythagorean Theorem, 492 radius and, 313–314 segments, 314 as term, use of, 69 tangent circles, 315 Tangent Conjecture, 313, 314 tangent ratio (tan), 620, 621–622, 624 tangent segments, 314 Tangent Segments Conjecture, |
314 Tangent Theorem, 699–700 tangram puzzle, 484 Taoism, 316 tatami, 386 technology applications, 52, 112, 131, 235, INDEX 761 263, 271, 283, 314, 317, 323, 339, 351, 423, 435, 498, 583, 607, 628, 630 exercises, 123, 145, 150, 160, 170, 180, 181, 186, 201, 259, 284, 316, 318, 323, 345, 382–383, 397, 427, 429, 436, 550, 585, 610, 645, 648, 650, 691, 710, 711 telecommunications, 112, 435, 498 temari balls, 720 temperature conversion, 210 term, nth, 106–108 Tessellating Quadrilaterals Conjecture, 385 Tessellating Triangles Conjecture, 384 tessellation (tiling) creation of, 21, 388, 389–391, 393–396 defined, 20 dual of, 382 glide reflection, 398–399 monohedral, 379–380, 384–385 nonperiodic, 388 with nonregular polygons, 384–385 regular, 380 rotation, 393–395 semiregular, 380–381 translation, 389–390 vertex arrangement, 380, 381 test problems, writing, 254 tetrahedron, 505 Thales of Miletus, 233, 583, 668 theodolite, 628 Theorem of Pythagoras. See Pythagorean Theorem theorem(s), 668 defined, 463 logical family tree of, 682–684 proving. See proof(s) See also specific theorems listed by name therefore, symbol for, 552 Thiebaud, Wayne, 514 thinking backward, 294 Third Angle Conjecture, 200–201 Third Angle Theorem, 682–684 30°-60°-90° triangle, 476–477 30°-60°-90° Triangle Conjecture, 476–477 Thomas, Calista, 647 Thompson, Rewi, 709 Three Midsegments Conjecture, 273 3-uniform tiling, 381 Three Worlds (Escher), 27 Tibet, 576 tiling. See tessellation transformation(s), 358 nonrigid, 358, 566–567, 578–580 762 INDEX rigid. See isometry trans |
itive property of congruence, 671 transitive property of equality, 670 transitive property of similarity, 706 translation, 358–359 and composition of isometries, 373–374, 376 defined, 358 direction of, 358 distance of, 358 tessellations by, 389–390 as type of isometry, 358 vector, 358 transversal line, 126 trapezium, 268 trapezoid(s) arch design and, 271 area of, 417–418 base angles of, 267 bases of, 267 defined, 62 diagonals of, 269, 699 height of, 417 isosceles, 268–269 linkages of, 283 midsegments of, 273, 274–275 proofs involving, 699 properties of, 268–269 Trapezoid Area Conjecture, 418 Trapezoid Consecutive Angles Conjecture, 268 Trapezoid Midsegment Conjecture, 275 tree diagrams, 78 triangle(s) acute, 60, 636, 641–642 adjacent interior angles of, 215–216 altitudes of, 154, 177, 401, 586 angle bisectors of, 176–179, 586, 587–588 area of, 411, 417, 454, 634–635 centroid of, 183–185, 189–190, 402 circumcenter. See circumcenter circumscribed, 71, 179 congruence of, 168–169, 219–222, 225–227, 230–231 constructing, 143, 168–169, 205 definitions of, 60–61 determining parts of, 168 drawing, 134 elliptic geometry and, 720 equiangular. See equilateral triangle(s) equilateral. See equilateral triangle(s) exterior angles of, 215–216 height of, 154, 634–635 incenter, 177–179 included angle, 219 included side, 219 inequalities, 213–216 inscribed, 71, 179 interior angles of, 215–216 isosceles. See isosceles triangle(s) medians of, 149, 183–185, 402, 586–587 midsegments of, 149, 273–274, 275 naming of, 54 obtuse, 60, 641–642 orthocenter. See orthocenter parallel lines and proportions of obtuse, 641–642 perpendicular bisectors of, 149, 176–178 points of concurrency of. See point(s) of concurrency proofs involving, 681– |
684, 686–688, 706–709 relationships of, 78 remote interior angles of, 215–216 right. See right triangle(s) scalene, 384 similarity of, 200–201, 572–574, 586–588 sum of angles of, 199–200 symbol for, 54 tessellations with, 379–381, 384, 394–395 vertex angle, 62, 242–243 Triangle Area Conjecture, 417 Triangle Exterior Angle Conjecture, 216 Triangle Inequality Conjecture, 214 Triangle Midsegment Conjecture, 274, 275 Triangle Sum Conjecture, 198–201 Triangle Sum Theorem, 681–682 triangular numbers, 115 triangular prism, 506 triangular pyramid, 506 triangulation, 229 trigonometry, 620 adjacent side, 620 cosine (cos), 621–622 graphs of functions, 654 inverse cosine (cos1), 624 inverse sine (sin1), 624 inverse tangent (tan1), 624 Law of Cosines, 641–643, 647 Law of Sines, 634–637, 647 opposite side, 620 and periodic phenomena, 654 problem solving with, 627, 647 ratios, 620–624 sine (sin), 621–622 tables and calculators for, 622–624, 654 tangent (tan), 620, 621–622, 624 unit circle and, 651–654 vectors and, 647 truncated pyramid, 685 Tsiga series (Vasarely), 3 Turkey, 379, 668 Twain, Mark, 59, 104 two-column proof, 655, 687–688 two-point perspective, 174–175 2-uniform tiling, 381 Tyson, Cicely, 157 U undecagon, 54 unit circle, 651–654 units area and, 413 nautical mile, 351 not stated, 31 volume and, 514 Using Your Algebra Skills Coordinate Proof, 712–717 Finding the Circumcenter, 329–330 Finding the Orthocenter and Centroid, 401–403 Midpoint, 36–37 Proportion and Reasoning, 560–561 Radical Expressions, 473–474 Slope, 133–134 Slopes of Parallel and Perpendicular Lines, 165–166 Solving Systems of Linear Equations, 285–286 Writing Linear Equations, 210–211 Uzbekistan, 60 V VA |
Theorem (Vertical Angles Theorem), 679–680 valid argument, 100, 102, 551 valid reasoning. See logic vanishing point(s), 172, 173, 174 Vasarely,Victor, 3, 13 vector(s) defined, 280 diagrams with, 280–281 resultant, 281 translation, 358 trigonometry with, 647 vector sum, 281 velocity and speed calculations, 134, 293, 302, 337, 338, 340, 344, 345, 351, 392, 483, 497, 660, 661 velocity vectors, 280–281 Venn diagram, 78 Venters, Diane, 56 Verblifa tin (Escher), 503 Verne, Jules, 337 vertex (vertices) of a cone, 508 consecutive, 54 defined, 38 naming angles by, 38 of a polygon, 54 of a polyhedron, 505 of a pyramid, 506 tessellation arrangement, 380, 381 vertex angle(s) bisector of, 242–243 of an isosceles triangle, 62, 242–243 of a kite, 266 Vertex Angle Bisector Conjecture, 242 vertex arrangement, 380, 381 vertical angles, 50, 121, 679–680 Vertical Angles Conjecture, 121–122, 129 Vertical Angles Theorem (VA Theorem), 679–680 vintas, 144 Vichy-Chamrod, Marie de, 142 Vietnam Veterans Memorial Wall, 130 volume of a cone, 522–524 of a cylinder, 515–517 defined, 514 displacement and density and, 535–536 of a hemisphere, 542–543 maximizing, 538 of a prism, 515–517 problems in, 531 proportion and, 593–594 of a pyramid, 522–524 of a sphere, 542–543 and surface area, relationship of, 599–602 units used to measure, 514 Vries, Jan Vredeman de, 172 W Walker, Mary Willis, 546 Wall Drawing #652 (LeWitt), 61 Warhol, Andy, 507 water and buoyancy, 537 and volume, 520, 537 Water Series (Greve), 306 Waterfall (Escher), 461 Weyl, Hermann, 358 Wick, Walter, 66 wigwams, 548 Wilcox, Ella Wheeler, 647 Wilde, Oscar, 462 Wiles, Andrew, 74 Williams, William T., |
79 woodworking, 34 work, 484 World Book Encyclopedia, 26 Wright, Frank Lloyd, 9 Wright, Steven, 514 writing test problems, 254 Y y-intercept, 210 yin-and-yang symbol, 316 Z zero product property of equality, 670 Zhoubi Suanjing, 502 zillij, 22 zoology and animal care, 15, 435, 536, 576, 694 I n d e x INDEX 763 Photo Credits Abbreviations: top (T), center (C), bottom (B), left (L), right (R). Cover Background image: Doug Wilson/Corbis; Construction image: Sonda Dawes/The Image Works; All other images: Ken Karp Photography. Front Matter v (T): Ken Karp Photography; v (C): Cheryl Fenton; v (B): Cheryl Fenton; vi (T): Ken Karp Photography; vii (T): Ken Karp Photography; vii (C): Courtesy, St. John’s Episcopal Church; vii (B): Hillary Turner; viii (T): Ken Karp Photography; viii (B): Corbis/Stockmarket; ix (T): Cheryl Fenton; ix (B): Cheryl Fenton; x (T): Ken Karp Photography; xi (T): Ken Karp Photography; xii (T): Ken Karp Photography; xii (B): Perry Collection/Photo by Cheryl Fenton; xiii: Cheryl Fenton. Chapter 0 1: Print Gallery, M. C. Escher, 1956/©2002 Cordon Art B.V.–Baarn– Holland. All rights reserved.; 2 (B): ©1993 Metropolitan Museum of Art, Bequest of Edward C. Moore, 1891 (91.1.)2064 2 (C): Cheryl Fenton; 2 (TL): NASA; 3 (BL): Christie’s Images; 3 (T): Tsiga I,II,III (1991),Victor Vasarely, Courtesy of the artist.; 4 (CR): Hillary Turner; 4 (TL): Cheryl Fenton; 4 (TR): Cheryl Fenton; 5 (B): ©Andy Goldsworthy, Courtesy of the artist and Galerie Lelong; 5 (C): Cheryl Fenton; 5 (CL): Cheryl Fenton; 5 (TC): Cheryl Fenton; 5 (TL): Cheryl Fenton; 6: Corbis; 7 (BL): Dave Bartruff/ |
Stock Boston; 7 (BR): Robert Frerck/Woodfin Camp & Associates; 7 (CL): Rex Butcher/Bruce Coleman Inc.; 7 (CR): Randy Juster; 9: Schumacher & Co./Frank Lloyd Wright Foundation; 10 (R): Sean Sprague/Stock Boston; 10 (TC): Christie’s Images/Corbis; 12: W. Metzen/Bruce Coleman Inc.; 13 (L): Hesitate, Bridget Riley/Tate Gallery, London/Art Resource, NY; 13 (R): Harlequin by Victor Vasarely, Courtesy of the artist.; 15 (C): National Tourist Office of Spain; 15 (T): Tim Davis/Photo Researchers; 16: Cheryl Fenton; 17: Snakes, M. C. Escher, 1969/©2002 Cordon Art B.V.– Baarn–Holland. All rights reserved.; 18: Will & Deni McIntyre/ Photo Researchers Inc.; 19: SEKI/PY XVIII (1978), Kunito Nagaoka/ Courtesy of the artist.; 19 (L): Cheryl Fenton; 19 (R): Cheryl Fenton; 20 (B): Corbis; 20 (T): Nathan Benn/Corbis; 22 (B): Ken Karp Photography; 22 (C): Peter Sanders Photography; 22 (TL): Peter Sanders Photography; 22 (TR): Peter Sanders Photography; 23: Photo Researchers Inc.; 24: Hot Blocks (1966–67) ©Edna Andrade, Philadelphia Museum of Art, Purchased by Philadelphia Foundation Fund; 25 (L): Comstock; 25 (R): Scala/Art Resource; 29 (T): George Lepp/Photo Researchers Inc. Chapter 1 27: Three Worlds, M. C. Escher, 1955/©2002 Cordon Art B.V.– Baarn–Holland. All rights reserved.; 28 (B): Spencer Grant/Photo Researchers Inc.; 28 (C): Cheryl Fenton; 28 (T): Hillary Turner; 29: By permission of Johnny Hart and Creators Syndicate, Inc.; 30: Bachman/Photo Researchers Inc.; 32: S. Craig/Bruce Coleman Inc.; 33 (R): Grafton Smith/Corbis Stock Market; 33 (L): Michael Daly/Corbis Stock Market; 34: Addison Geary/Stock Boston; 35: Bob Stovall/Bruce Coleman Inc.; 36: Archivo Iconografico, S. A./Corbis; 38 (TL): |
Bruce Coleman Inc.; 38 (TR): David Leah/Getty Images; 39 (B): Hillary Turner; 39 (BR): Cheryl Fenton; 39 (BR): Comstock; 39 (T): Ken Karp Photography; 41: Pool & Billiard Magazine; 47: Illustration by John Tenniel; 48: Osentoski & Zoda/Envision; 49: Christie’s Images; 50: Corbis; 51: Hillary Turner; 54: Cheryl Fenton; 55: Ira Lipsky/International Stock Photography; 56: Quilt by Diane Venters/More Mathematical Quilts; 56 (C): Cheryl Fenton; 56 (TCL): Hillary Turner; 56 (TCR): Hillary Turner; 56 (TL): Hillary Turner; 56 (TR): Hillary Turner; 59: Spencer Swanger/Tom Stack & Associates; 60: Gerard Degeorge/Corbis; 61: Sol LeWitt—Wall Drawing #652—On three walls, continuous forms with color ink washes superimposed, Color in wash. Collection: Indianapolis Museum of Art, Indianapolis, IN. September, 1990. Courtesy of the Artist.; 62 (C): Michael Moxter/Photo Researchers Inc.; 62 (L): Larry Brownstein/Rainbow; 62 (R): Stefano Amantini/Bruce Coleman Inc.; 63 (B): Cheryl Fenton; 63 (T): Cheryl Fenton; 65: Friedensreich ©Erich Lessing/Art Resource, NY; 66: From WALTER WICK’S OPTICAL TRICKS. Published by Cartwheel books, a division of Scholastic Inc. ©1998 by Walter Wick. Reprinted by permission.; 67 (BL): Joel Tribhout/ Agence Vandystadt/Getty Images; 67 (BR): Terry Eggers/Corbis Stock Market; 67 (T): By permission of Johnny Hart and Creators Syndicate Inc.; 68 (L): Cheryl Fenton; 68 (R): Getty Images; 70 (CL): Corbis; 70 (CR): Alfred Pasieka/Photo Researchers Inc.; 70 (T): Corbis; 73: Bookplate for Albert Ernst Bosman, M. C. Escher, 1946/©2002 Cordon Art B.V.–Baarn–Holland. All rights reserved.; 74 (BC): Stock Montage, Inc.; 74 (BL): Stock Montage, Inc.; 74 (BR): AP/Wide World; 79: William Thomas Williams, |
“DO YOU THINK A IS B,” Acrylic on Canvas, 1975–77, Fisk University Galleries, Nashville, Tennessee; 80 (C): Cheryl Fenton; 80 (L): Cheryl Fenton; 80 (R): Cheryl Fenton; 81: Cheryl Fenton; 82: Cheryl Fenton; 83: Courtesy of Kazumata Yamashita, Architect; 84 (B): Ken Karp Photography; 84 (C): Mike Yamashita/Woodfin Camp & Associates; 85: T. Kitchin/Tom Stack & Associates; 86 (C): Cheryl Fenton; 86 (T): Ken Karp Photography; 88: Paul Steel/Corbis-Stock Market; 90: Cheryl Fenton. Chapter 2 93: Hand with Reflecting Sphere (Self-Portrait in Spherical Mirror), M. C. Escher/©2002 Cordon Art B.V.–Baarn–Holland. All rights reserved.; 94 (B): Barry Rosenthal/FPG; 94 (T): Ken Karp Photography; 95: Andrew McClenaghan/Photo Researchers Inc.; 100: Bob Daemmrich/Stock Boston; 101: California Institute of Technology and Carnegie Institution of Washington; 104: NASA; 106: Drabble reprinted by permission of United Feature Syndicate, Inc.; 112 (L): National Science Foundation Network; 112 (R): Hank Morgan/Photo Researchers Inc.; 113: Hillary Turner; 115: Cheryl Fenton; 118 (B): Ken Karp Photography; 118 (T): Culver Pictures; 121: Ken Karp Photography; 126 (B): Ken Karp Photography; 126 (T): Alex MacLean/Landslides; 128: Hillary Turner; 130: James Blank/Bruce Coleman Inc.; 134 (L): Photo Researchers; 134 (R): Mark Gibson/Index Stock; 135 (B): Ted Scott/Fotofile, Ltd.; 135 (C): Ken Karp Photography; 137: Art Matrix. Chapter 3 141: Drawing Hand, M. C. Escher, 1948/©2002 Cordon Art B.V.– Baarn–Holland. All rights reserved.; 142 (L): Hillary Turner; 142 (R): Hillary Turner; 142 (T): Bettmann/Corbis; 143 (L): Hillary Turner; 143 (R): Hillary Turner; 144: Travel Ink/Corbis; 154: Overseas Highway by Crawford Ralston, 1939 by Crawford Ralston/Art Resource, NY; |
156: Rick Strange/Picture Cube; 159: Corbis; 172 (BL): Ken Karp Photography; 172 (BR): Dover Publications; 172 (T): Greg Vaughn/Tom Stack & Associates; 174: Art Resource; 175: Timothy Eagan/Woodfin Camp & Associates; 176: Ken Karp Photography; 177: Rounds and Triangles by Rudolf Bauer /Christie’s Images; 180: Corbis; 185 (B): Corbis; 185 (T): Ken Karp Photography; 186: Keith Gunnar/Bruce Coleman Inc.; 189: Ken Karp Photography; 191: Victoria & Albert Museum, London/Art Resource, NY. Chapter 4 197: Symmetry Drawing E103, M. C. Escher,1959/©2002 Cordon Art B.V.–Baarn–Holland. All rights reserved.; 198 (B): Courtesy, St. John’s Presbyterian Church; 198 (C): Jim Corwin/Stock Boston; 198 (TL): Cheryl Fenton; 198 (TR): Cheryl Fenton; 199: Ken Karp Photography; 202: The Far Side® by Gary Larson ©1987 FarWorks, Inc. All rights reserved. Used with permission.; 203: Hillary Turner; PHOTO CREDITS 765 204 (L): David L. Brown/Picture Cube; 204 (R): Joe Sohm/The Image Works; 207: Art Stein/Photo Researchers Inc.; 213: Robert Frerck/Woodfin Camp & Associates; 217: Judy March/Photo Researchers Inc.; 220: Adamsmith Productions/Corbis; 222: Carolyn Schaefer/Gamma Liaison; 227: Cheryl Fenton; 234 (B): Ken Karp Photography; 234 (T): AP/Wide World; 240: Ken Karp Photography; 241: Eric Schweikardt/The Image Bank; 247: Archivo Iconografico, SA/Corbis; 249 (T): Ken Karp Photography; 251: Ken Karp Photography; 252 (L): Cheryl Fenton; 252 (R): Cheryl Fenton; 253: Cheryl Fenton. Chapter 5 255: Still Life and Street, M. C. Escher, 1967–68/©2002 Cordon Art B.V.–Baarn–Holland. All rights reserved.; 256: Ken Karp Photography; 256 (T): Cheryl Fenton; 260: Emerging Order by Hannah Hoch/Christie’s Images/Corbis; 263: Richard Megna/Fundamental |
Photographs; 265: Courtesy of the artist, Teresa Archuleta-Sagel; 266: Steve Dunwell/The Image Bank; 268 (C): Lindsay Hebberd/Woodfin Camp & Associates; 268 (T): Photo Researchers Inc.; 271: Cheryl Fenton; 278: Malcolm S. Kirk/Peter Arnold Inc.; 284: ©Paula Nadelstern/Photo by Bobby Hansson; 286: Corbis; 289: Ken Karp Photography; 290: Cheryl Fenton; 291: Bill Varie/The Image Bank; 292: Bill Varie/The Image Bank; 292: Alex MacLean/Landslides; 298: Cheryl Fenton; 299: Red and Blue Puzzle (1994) Mabry Benson/Photo by Carlberg Jones; 302: Ken Karp Photography; 303: Boy With Birds by David C. Driskell/Collection of Mr. and Mrs. David C. Driskell. Chapter 6 305: Curl-Up, M. C. Escher,1951/©2002 Cordon Art B.V.–Baarn– Holland. All rights reserved.; 306 (B): Stephen Saks/Photo Researchers, Inc.; 306 (C): Corbis; 308: Tom Sanders/Corbis Stock Market; 311: NASA; 312: Tony Freeman/PhotoEdit; 313 (L): Stephen A. Smith; 313 (R): Mark Burnett/Stock Boston; 314: NASA; 315: Gray Mortimore/Getty Images; 316 (L): Corbis; 316 (L): Jim Zuckerman/Corbis; 319: Xinhua/Gamma Liaison; 328 (L): Cheryl Fenton; 328 (R): Cheryl Fenton; 331 (B): Cheryl Fenton; 331 (C): Ken Karp Photography; 332: Hillary Turner; 334 (B): Don Mason/Corbis Stock Market; 334 (C): David R. Frazier/Photo Researchers Inc.; 334 (T): Ken Karp Photography; 337: Photofest; 339: Calvin and Hobbes ©Watterson. Reprinted with permission of UNIVERSAL PRESS SYNDICATE. All rights reserved.; 340: Guido Cozzi/Bruce Coleman Inc.; 341: Grant Heilman Photography; 344 (B): Nick Gunderson/Corbis; 344 (T): Charles Feil/Stock Boston; 345 (B): Bob Daemmrich/ Stock Boston; 345 (T): Cathedrale de Reims; 346: Gamma Lia |
ison; 349 (B): Ken Karp Photography; 349 (T): Cheryl Fenton; 351: Ken Karp Photography; 352: The Far Side® by Gary Larson ©1990 FarWorks, Inc. All rights reserved. Used with permission.; 353: David Malin/Anglo-Australian Telescope Board; 355: Private collection, Berkeley, California/Ceramist, Diana Hall. Chapter 7 357: Magic Mirror, M. C. Escher,1946/©2002 Cordon Art B.V.– Baarn–Holland. All rights reserved.; 358: Giraudon/Art Resource; 358 (B): Ken Karp Photography; 360: Corbis; 361: Grant Heilman Photography; 363 (BC): Denver Museum of Natural History; 363 (BL): Michael Lustbader/Photo Researchers; 363 (BR): Richard Cummins/Corbis; 363 (TL): Phil Cole/Getty Images; 363 (TR): Corbis; 364: Oxfam; 369: Corbis; 372: By Holland ©1976 Punch Cartoon Library; 373: Gina Minielli/Corbis; 378: Ken Karp Photography; 379 (L): W. Treat Davison/Photo Researchers; 379 (R): Ancient Art and Architecture Collection; 381: SpringerVerlag; 385: Courtesy Doris Schattschneider; 386: Lucy Birmingham/ Photo Researchers Inc.; 388 (B): Carleton College/photo by Hilary N. Bullock; 388 (T): Ashton, Ragget, McDougall Architects; 389 (C): Symmetry Drawing E105, M. C. Escher 1960/©2002 Cordon Art B.V.–Baarn–Holland. All rights reserved.; 389 (T): Brickwork, Alhambra, M. C. Escher/©2002 Cordon Art B.V.–Baarn–Holland. All rights reserved.; 393: Symmetry Drawing E25, M. C. Escher, 1939/©2002 Cordon Art B.V.–Baarn–Holland. All rights reserved.; 394: Reptiles, M. C. Escher, 1943/©2002 Cordon Art B.V.–Baarn–Holland. All rights reserved.; 395 (T): Symmetry Drawing E99, M. C. Escher, 1954/©2002 Cordon Art B.V.–Baarn– Holland. All rights reserved.; 396 (L): Inter |
woven patterns–V– structure 17 by Rinus Roelofs/Courtesy of the artist & ©2002 Artist Rights Society (ARS), New York/Beeldrecht, Amsterdam; 396 (R): Impossible structures–III–structure 24 by Rinus Roelofs/ Courtesy of the artist & ©2002 Artist Rights Society (ARS), New York/Beeldrecht, Amsterdam; 398 (B): Horseman sketch, M. C. Escher/©2002 Cordon Art B.V.–Baarn–Holland. All rights reserved.; 398 (T): Horseman, M. C. Escher, 1946/©2002 Cordon Art B.V.– Baarn–Holland. All rights reserved.; 399: Symmetry Drawing E108, M. C. Escher, 1967/©2002 Cordon Art B.V.–Baarn–Holland. All rights reserved.; 404: King by Minne Evans/Corbis & Luise Ross Gallery; 405 (C): Paul Schermeister/Corbis; 405 (TR): Comstock; 406: Ken Karp Photography; 407 (B): Starbuck Goldner; 407 (C): Day and Night, M. C. Escher, 1938/©2002 Cordon Art B.V.–Baarn– Holland. All rights reserved. Chapter 8 409: Square Limit, M. C. Escher, 1964/©2002 Cordon Art B.V.– Baarn–Holland. All rights reserved.; 410: Spencer Grant/Stock Boston; 414 (B): Karl Weatherly/Corbis; 414 (T): Courtesy Naoko Hirakura Associates; 421: Ken Karp Photography; 422: Joseph Nettis/Photo Researchers Inc.; 423 (B): NASA; 423 (C): Ken Karp Photography; 424: Phil Schermeister/Corbis; 428: Cheryl Fenton; 434: Cheryl Fenton; 435 (B): Reuters NewMedia Inc./Corbis; 435 (C): Georg Gerster/Photo Researchers Inc.; 437: Shahn Kermani/ Gamma Liaison; 440: Eye Ubiquitous/Corbis; 442: Ken Karp Photography; 444: Cheryl Fenton; 445 (BL): Stefano Amantini/ Bruce Coleman Inc.; 445 (BR): Alain Benainous/Gamma Liaison; 445 (CB): Joan Iaconetti/Bruce Coleman Inc.; 445 (CT): Bill Bachmann/The |
Image Works; 446: Greg Pease/Corbis; 447 (L): Sonda Dawes/The Image Works; 447 (R): John Mead/ Photo Researchers Inc.; 449: Library of Congress; 451: Bryn Campbell/Getty Images; 452: Douglas Peebles; 458: Ken Karp Photography. Chapter 9 461: Waterfall, M. C. Escher, 1961/©2002 Cordon Art B.V.–Baarn– Holland. All rights reserved.; 462: FUNKY WINKERBEAN by Batiuk. Reprinted with special permission of North American Syndicate; 463: Corbis; 463 (L): Corbis; 464: Will & Deni McIntyre/ Photo Researchers Inc.; 467 (C): Corbis; 467 (T): Ken Karp Photography; 468: Ken Karp Photography; 469: Rare Book and Manuscript Library/Columbia University; 470: John Colett/Stock Boston; 472: Christie’s Images/©2002 Sol LeWitt/Artists Rights Society (ARS), New York; 475: Turner Entertainment ©1939. All rights reserved.; 482: FUNKY WINKERBEAN by Batiuk. Reprinted with special permission of North American Syndicate; 483: Bob Daemmrich/The Image Works; 484 (B): Private collection, Berkeley, California/Photo by Cheryl Fenton; 484 (T): Ken Karp Photography; 490: David Muench/Corbis; 494: Bruno Joachim/ Gamma Liaison; 495: Ken Karp Photography; 498: Ken Karp Photography. Chapter 10 503: Verblifa tin, M. C. Escher, 1963/©2002 Cordon Art B.V.–Baarn– Holland. All rights reserved.; 504 (BL): J. Bernholc, etal./North Carolina State/Photo Researchers Inc.; 504 (CB): Ken Eward/Photo Researchers Inc.; 504 (CT): Runk-Schoenberger/Grant Heilman Photography; 504 (CT): Ken Eward/Photo Researchers; 504 (L): Archivo Iconografico, S. A. /Corbis; 504 (T): Cheryl Fenton; 505: Esaias Baitel/Gamma Liaison; 506: Cheryl Fenton; 507: Burstein Collection/Corbis/© Andy Warhol/Art |
ists Rights Society (ARS), New York; 510: ©1996 C. Herscovici, Brussels, Artists Rights Society (ARS) NY/Photo courtesy Minneapolis Institute of 766 PHOTO CREDITS Art; 512: Ken Karp Photography; 514: Ken Karp Photography; 514: Christie’s Images/Corbis; 516: Hillary Turner; 519 (B): Jeff Tinsly/Names Project; 519 (C): Larry Lee Photography/ Corbis; 520 (B): Thomas Kitchin/Tom Stack & Associates; 520 (C): Eye Ubiquitous/Corbis; 521 (B): Cheryl Fenton; 521 (C): Cheryl Fenton; 522: Ken Karp Photography; 525: Masao Hayashi/Photo Researchers Inc.; 527: Charles Lenars/Corbis; 528 (B): Ken Karp Photography; 528 (T): Fitzwilliam Museum, Cambridge; 532 (T): David Sutherland/Stone/Getty Images; 533: Parson’s School of Design; 537: David Morris/Gamma Liaison; 539 (B): C. J. Allen/Stock Boston; 539 (T): Ken Karp Photography; 540: David Hockney, Sunday Morning Mayflower Hotel, N.Y., Nov. 28, 1982/Photographic collage, ED: 20/50 × 77 ©David Hockney; 541: Ken Karp Photography; 542 (B): Ken Karp Photography; 544 (B): The Far Side® by Gary Larson ©1986 FarWorks, Inc. All rights reserved. Used with permission.; 544 (T): Culver Pictures/ Picture Quest; 546 (C): John Cooke/Comstock; 546 (T): NASA; 548: Larry Lefevre/Grant Heilman Photography; 549: Arte & Immagini/Corbis; 551 (L): Ken Karp Photography; 551 (T): Illustration by Sidney Paget from The Strand Magazine, 1892; 555: Piranha Club by B. Grace. Reprinted with special permission of King Features Syndicate; 558: Douglas Peebles/Corbis. Chapter 11 559: Path of Life I, M. C. Escher, 1958/©2002 Cordon Art B.V.– Baarn–Holland. All rights reserved.; 564 (R): J. Greenberg/ |
The Image Works; 564 (R): Chromosohn/Photo Researchers Inc.; 567 (B): National Geographic Society; 567 (C): Giaudon/Art Resource, NY; 567 (T): Erich Lessing/Art Resource, NY; 569: John Elk III/Stock Boston; 570: Daniel Sheehan/Black Star Publishing/PictureQuest; 571 (C): Ken Karp Photography; 571 (T): Michael S.Yamashita/Corbis; 575: Patricia Lanza/Bruce Coleman Inc.; 576 (B): Thomas Dove/Douglas Peebles; 576 (L): Rossi & Rossi; 576 (R): Rossi & Rossi; 577: Dave Bartruff/Corbis; 583: Ken Karp Photography; 586: Ken Karp Photography; 592: Alon Reininger/Contact Press Images/PictureQuest; 594: Ken Karp Photography; 595: David Fraser/Photo Researchers Inc.; 597 (B): Dr. Paul A. Zahl/Photo Researchers Inc.; 597 (T): La Petite Chatelaine, Version a La Natte Courbe by Camille Claudel/Christie’s Images/© 2002 Artists Rights Society (ARS), New York/ADAGP, Paris; 599: Ken Karp Photography; 600 (B): ©1996 Demart Pro Arte Geneva/© Salvador Dali, Gala-Salvador Dali Foundation/ARS NY; 600 (T): Corbis; 601 (C): Kurt Krieger/Corbis; 601 (T):Bettmann/ Corbis; 607: Charles Feil/Stock Boston; 611 (B): Ken Karp Photography; 612: Alice in Wonderland; 615 (B): Robert Holmes/ Corbis; 615 (T): Dan McCoy/Rainbow; 616 (L): Chris Lisle/Corbis; 616 (R): Alex Webb/Magnum; 617: Underwood & Underwood/ Corbis; 618: NASA. Chapter 12 619: Belvedere, M. C. Escher, 1958/©2002 Cordon Art B.V.–Baarn– Holland. All rights reserved.; 620: Bettmann/Corbis; 621 (B): Perry Collection; 621 (T): Gary Braasch/Woodfin Camp & Associates; 623: Cheryl Fenton; 627: |
Ken Karp Photography; 628 (B): Ecoscene/ Corbis; 628 (C): H. Reinhard/Photo Researchers; 629: Breezing Up by Winslow Homer/Photo by Francis G. Mayer/Corbis; 630: Dennis Marsico/Corbis; 631 (L): Wendell Metzen/Bruce Coleman Inc.; 631 (R): Corbis; 632 (C): Ken Karp Photography; 633: Greg Rynders; 638 (C): Courtesy of Association for the Preservation of Virginia Antiquities; 639 (B): Stevan Stefanovic/Photo Researchers Inc.; 639 (T): Steve Owlett/Bruce Coleman Inc.; 644 (B): Grant Heilman Photography; 646: Photograph by Hiroshi Umeoka; 647: Sonda Dawes/The Image Works; 649: James A. Sugar/Black Star Publishing/Picture Quest; 651: Angelo Hornak/Corbis; 652: Walter Hodges/Corbis; 654: Courtesy California Academy of Sciences; 655: Ken Karp Photography; 660: © Marty Sohl; 661: B. Christensen/Stock Boston. Chapter 13 667: Another World (Other World), M. C. Escher/©2002 Cordon Art B.V.–Baarn–Holland. All rights reserved.; 668: Springer-Verlag; 668: Araldo de Luca/CORBIS; 669 (C): Ken Karp Photography; 669 (R): Ken Karp Photography; 670: Charles & Josette Lenars/ Corbis; 671: Art Resource; 675: Library of Congress; 679: ©1977 by Sidney Harris, American Scientist Magazine; 682: Ken Karp Photography; 684: Ken Karp Photography; 703: M. Dillon/Corbis; 709: Paul A. Souders/Corbis; 718 (L): Springer-Verlag; 718 (R): Ken Karp Photography; 720: Collection of Suzanne Summer/Cheryl Fenton Photography PHOTO CREDITS 767es? Round to the nearest thousandth gram. Record your answer and fill in the bubbles on your answer document. Be sure to use the correct place value. DAY 5 Which conjecture about polygons is NOT true? The area of a parallelogram is the product of its base and height. A rhombus has four right angles. A square |
has four congruent sides. A trapezoid has exactly one pair of parallel sides. Which Pythagorean triple would be most helpful in finding the value of a? 3-4-5 5-12-14 8-15-17 7-24-25 DAY 4 Natalia plans to install glass doors across the front of her square fireplace opening and then seal the perimeter of the opening with a special caulk that can sustain high temperatures. What is the perimeter of the opening? Record your answer and fill in the bubbles on your answer document. Be sure to use the correct place value. Countdown to TAKS TX21 TX21 ������������������������������������������������ Countdown to TAKS WEEK 19 DAY 1 Which two line segments are congruent? ̶̶ AB and ̶̶ CE and ̶̶̶ GH and ̶̶ CD and ̶̶ DF ̶̶̶ GH ̶̶ AB ̶̶ DE DAY 2 DAY 3 Based on the table, which algebraic expression best represents the number of triangles formed by drawing all of the diagonals from one vertex in a polygon with n sides? At a certain time of the day, a 24-foot tree casts an 18-foot shadow. How long is the shadow cast by a 4-foot mailbox at the same time of day? 3 1 4 2 5 3 8 6 No. of sides No. of triangles formed n 2n - 1 n - 2 n + 2 _ 2 ����� ����� ���� Record your answer and fill in the bubbles on your answer document. Be sure to use the correct place value. DAY 4 DAY 5 A school increases the width of its rectangular playground from 25 meters to 40 meters and the length from 45 meters to 60 meters. By how much does the perimeter of the playground increase? Trey is using triangular tiles to floor his bathroom. What is x? 30 meters 60 meters 200 meters 225 meters TX22 TX22 Countdown to TAKS Record your answer and fill in the bubbles on your answer document. Be sure to use the correct place value. ������������������������������ Countdown to TAKS WEEK 20 DAY 1 The figure shows the measure of each interior angle for several regular polygons. Which algebraic expression best represents the measure of an interior angle of a regular polygon with n sides? (n - 2) 180 __ n 360n _ n + 2 (n - 2) 180 180n _ 2 DAY 2 DAY 3 Which coordinates represent a vertex of |
the hexagon? The two triangles in the figure are similar. What is the length of ̶̶̶ MN? (0, 2) (4, -2) (3, 2) (-2, 2) Record your answer and fill in the bubbles on your answer document. Be sure to use the correct place value. DAY 4 DAY 5 Two regular pentagons have perimeters of 30 and 75 respectively. What scale factor relates the smaller figure to the larger one? Alissa is painting a diagonal line across a square tile. What is the length of the line in centimeters? Round to the nearest thousandth of a centimeter. 1 : 2.5 1 : 6 1 : 15 1 : 21 Record your answer and fill in the bubbles on your answer document. Be sure to use the correct place value. Countdown to TAKS TX23 TX23 ���������������������������������������� Countdown to TAKS WEEK 21 DAY 1 The table lists the measure of an exterior angle for the given regular polygon. Which expression best represents the measure of an exterior angle of a regular polygon with n sides? Figure Quadrilateral Pentagon Decagon Exterior angle 90° 72° 36° 360 _ n - 2 360 + n _ 2 + n 360n 360 _ n DAY 2 DAY 3 When y = 65, x = 8. If y varies directly with x, what is y when x equals 15? Record your answer and fill in the bubbles on your answer document. Be sure to use the correct place value. A word game uses a bag of 80 tiles. Forty of the tiles have a consonant on them, and the remaining 40 have a vowel: A, E, I, O, or U. There is an equal number of each vowel tile. What is the probability as a percent that Shelly selects an A tile and then a U tile from the bag without replacement? Round to the nearest hundredth of a percent. Record your answer and fill in the bubbles on your answer document. Be sure to use the correct place value. DAY 4 DAY 5 Which equation best describes the line containing the hypotenuse of this triangle? ̶̶ AB. What are the coordinates of the The center of circle C is the midpoint of midpoint TX24 TX24 Countdown to TAKS (0, 4) (1, 4) (2, 4) (3, 3) ���������������� Countdown to TAKS WEEK 22 DAY 1 If this pattern is continued, how many shaded triangles |
will there be in the fourth element of the pattern? 9 13 27 40 DAY 2 DAY 3 What is the slope of the line? A delivery truck travels 13.5 mi east and then 18 mi north. How far in miles is the truck from its starting point Record your answer and fill in the bubbles on your answer document. Be sure to use the correct place value. DAY 4 DAY 5 What are the side lengths of the triangle? 3, 4, and 5 2, 3, and 5 3, 3, and 3 3, 3, and 3 √ 2 An 18-foot ladder reaches the top of a building when placed at an angle of 45° with the horizontal. What is the approximate height of the building in feet? Round to the nearest tenth of a foot. Record your answer and fill in the bubbles on your answer document. Be sure to use the correct place value. Countdown to TAKS TX25 TX25 ���������������������������� Countdown to TAKS WEEK 23 DAY 1 DAY 2 △RST is a 30°-60°-90° triangle. What is the y-coordinate of R if a = -5 and c = -2? What is x if y is 12.8 and z is 16 in the right triangle below? 3.2 4.0 9.6 12. DAY 3 How does the slope of the hypotenuse of △ABC compare with that of △DBC? They have the same value and sign. They have opposite signs. One is a multiple of the other. They are reciprocals. DAY 4 DAY 5 How many sides does a regular polygon have if each interior angle measures 120°? Record your answer and fill in the bubbles on your answer document. Be sure to use the correct place value. A piñata in the shape of a basketball is filled with treats for a game during Hanj’s birthday party. If the diameter of the piñata is 7 inches, what is the volume of the piñata in cubic inches? Round to the nearest tenth. Record your answer and fill in the bubbles on your answer document. Be sure to use the correct place value. TX26 TX26 Countdown to TAKS ������������������������������������� Countdown to TAKS WEEK 24 DAY 1 Quadrilaterals ABCD and WXYZ are similar. What is XY? 3.5 21 24.5 35 DAY 2 DAY 3 The volume of a square pyramid is 108 |
cubic millimeters. What is the height of the pyramid in millimeters if one side on the base is 4.5 millimeters? Record your answer and fill in the bubbles on your answer document. Be sure to use the correct place value. What is the value of x in the regular pentagon below? 54° 90° 108° 180° DAY 4 DAY 5 What is the second term in a proportion in which the first, third, and fourth terms are 3, 9, and 12, respectively? Record your answer and fill in the bubbles on your answer document. Be sure to use the correct place value. The endpoints of a segment are Q (-2, 6) and R (5, -4). What is the length of the segment to the nearest tenth? 3.6 units 4.1 units 8.5 units 12.2 units Countdown to TAKS TX27 TX27 ���������������������� Texas Friendship Texas Essential Knowledge and Skills for Geometry a Basic understandings. 1 Foundation concepts for high school mathematics. As presented in Grades K–8, the basic understandings of number, operation, and quantitative reasoning; patterns, relationships, and algebraic thinking; geometry; measurement; and probability and statistics are essential foundations for all work in high school mathematics. Students continue to build on this foundation as they expand their understanding through other mathematical experiences. 2 Geometric thinking and spatial reasoning. Spatial reasoning plays a critical role in geometry; geometric figures provide powerful ways to represent mathematical situations and to express generalizations about space and spatial relationships. Students use geometric thinking to understand mathematical concepts and the relationships among them. 3 Geometric figures and their properties. Geometry consists of the study of geometric figures of zero, one, two, and three dimensions and the relationships among them. Students study properties and relationships having to do with size, shape, location, direction, and orientation of these figures. The State Capitol in Austin Texas wildflowers Houston skyline TX28 The state bird is the Mockingbird. Texas Essential Knowledge and Skills The Bluebonnet is the state flower. 4 The relationship between geometry, other mathematics, and other disciplines. Geometry can be used to model and represent many mathematical and real-world situations. Students perceive the connection between geometry and the real and mathematical worlds and use geometric ideas, relationships, and properties to solve problems. 5 Tools for geometric thinking. Techniques Statue of a Texas longhorn for working with spatial figures and their properties are essential in understanding underlying relationships. Students use a variety of |
representations (concrete, pictorial, numerical, symbolic, graphical, and verbal), tools, and technology (including, but not limited to, calculators with graphing capabilities, data collection devices, and computers) to solve meaningful problems by representing and transforming figures and analyzing relationships. 6 Underlying mathematical processes. Many processes underlie all content areas in mathematics. As they do mathematics, students continually use problemsolving, language and communication, connections within and outside mathematics, and reasoning (justification and proof ). Students also use multiple representations, technology, applications and modeling, and numerical fluency in problem solving contexts. TX29 Texas Friendship B recognize the historical development of geometric systems and know mathematics is developed for a variety of purposes; and C compare and contrast the structures and implications of Euclidean and non-Euclidean geometries. G.2 Geometric structure. The student analyzes geometric relationships in order to make and verify conjectures. The student is expected to: A use constructions to explore attributes of geometric figures and to make conjectures about geometric relationships; and b Knowledge and skills. G.1 Geometric structure. The student understands the structure of, and relationships within, an axiomatic system. The student is expected to: A develop an awareness of the structure of a mathematical system, connecting definitions, postulates, logical reasoning, and theorems; Big Bend National Park TX30 Texas Essential Knowledge and Skills B make conjectures about angles, lines, polygons, circles, and three-dimensional figures and determine the validity of the conjectures, choosing from a variety of approaches such as coordinate, transformational, or axiomatic. G.3 Geometric structure. The student applies logical reasoning to justify and prove mathematical statements. C use logical reasoning to prove statements are true and find counter examples to disprove statements that are false; The student is expected to: A determine the validity of a conditional statement, its converse, inverse, and contrapositive; D use inductive reasoning to formulate a conjecture; and E use deductive reasoning to prove a statement. B construct and justify statements about geometric figures and their properties; TX31 Texas Friendship West Texas TX32 G.4 Geometric structure. The student uses a variety of representations to describe geometric relationships and solve problems. The student is expected to: A select an appropriate representation (concrete, pictorial, graphical, verbal, or symbolic) in order to solve problems. G.5 Geometric patterns. The student uses a variety |
of representations to describe geometric relationships and solve problems. The student is expected to: A use numeric and geometric patterns to develop algebraic expressions representing geometric properties; B use numeric and geometric patterns to make generalizations about geometric properties, including properties of polygons, ratios in similar figures and solids, and angle relationships in polygons and circles; Texas Essential Knowledge and Skills C use properties of B use nets to represent The student is expected to: transformations and their compositions to make connections between mathematics and the real world, such as tessellations; and D identify and apply patterns from right triangles to solve meaningful problems, including special right triangles (45-45-90 and 30-60-90) and triangles whose sides are Pythagorean triples. G.6 Dimensionality and the geometry of location. The student analyzes the relationship between three-dimensional geometric figures and related two-dimensional representations and uses these representations to solve problems. The student is expected to: A describe and draw the intersection of a given plane with various three-dimensional geometric figures; and construct three-dimensional geometric figures; and C use orthographic and isometric views of three-dimensional geometric figures to represent and construct three-dimensional geometric figures and solve problems. G.7 Dimensionality and the geometry of location. The student understands that coordinate systems provide convenient and efficient ways of representing geometric figures and uses them accordingly. A use one- and two-dimensional coordinate systems to represent points, lines, rays, line segments, and figures; B use slopes and equations of lines to investigate geometric relationships, including parallel lines, perpendicular lines, and special segments of triangles and other polygons; and C derive and use formulas involving length, slope, and midpoint. SeaWorld TX33 G.10 Congruence and the geometry of size. The student applies the concept of congruence to justify properties of figures and solve problems. The student is expected to: A use congruence transformations to make conjectures and justify properties of geometric figures including figures represented on a coordinate plane; and B justify and apply triangle congruence relationships. Texas Friendship G.9 Congruence and the geometry of size. The student analyzes properties and describes relationships in geometric figures. The student is expected to: A formulate and test conjectures about the properties of parallel and perpendicular lines based on explorations and concrete models; B formulate and test conjectures about the properties and attributes of polygons and their component parts based on explorations and concrete models; C formulate and test conjectures about the properties |
and attributes of circles and the lines that intersect them based on explorations and concrete models; and D analyze the characteristics of polyhedra and other three-dimensional figures and their component parts based on explorations and concrete models. G.8 Congruence and the geometry of size. The student uses tools to determine measurements of geometric figures and extends measurement concepts to find perimeter, area, and volume in problem situations. The student is expected to: A find areas of regular polygons, circles, and composite figures; B find areas of sectors and arc lengths of circles using proportional reasoning; C derive, extend, and use the Pythagorean Theorem; and D find surface areas and volumes of prisms, pyramids, spheres, cones, cylinders, and composites of these figures in problem situations. TX34 Texas Essential Knowledge and Skills Texas State Fair G.11 Similarity and the geometry of shape. The student applies the concepts of similarity to justify properties of figures and solve problems. The student is expected to: A use and extend similarity properties and transformations to explore and justify conjectures about geometric figures; B use ratios to solve problems involving similar figures; C develop, apply, and justify triangle similarity relationships, such as right triangle ratios, trigonometric ratios, and Pythagorean triples using a variety of methods; and D describe the effect on perimeter, area, and volume when one or more dimensions of a figure are changed and apply this idea in solving problems. TX35 Foundations for Geometry KEYWORD: MG7 TOC ARE YOU READY TEKS G.7.A G.2.A G.3.B G.3.B G.1.A Euclidean and Construction Tools 1-1 Understanding Points, Lines, and Planes................... 6 Explore Properties Associated with Points....... 12 1-2 Measuring and Constructing Segments................... 13 1-3 Measuring and Constructing Angles...................... 20 1-4 Pairs of Angles......................................... 28 MULTI-STEP TAK |
S PREP................................... 34 READY TO GO ON? QUIZ................................. 35 Coordinate and Transformation Tools G.8.A 1-5 Using Formulas in Geometry............................. 36 G.1.A G.5.C G.2.A On Track for TAKS: Algebra Graphing in the Coordinate Plane...................... 42 1-6 Midpoint and Distance in the Coordinate Plane........... 43 1-7 Transformations in the Coordinate Plane.................. 50 Explore Transformations......................... 56 MULTI-STEP TAKS PREP................................... 58 READY TO GO ON? QUIZ................................. 59 Study Guide: Preview Reading and Writing Math................................. 5 Study Guide: Review..................................... 60 Chapter Test............................................ 64 Tools for Success Reading Math 5 Writing Math 10, 18, 26, 33, 40, 48, 54 Vocabulary |
3, 4, 9, 17, 24, 31, 38, 47, 53, 60 Know-It Notes 6, 7, 8, 13, 14, 16, 20, 21, 22, 24, 28, 29, 31, 36, 37, 43, 44, 45, 46, 50, 52 Test Prep Exercises 11, 19, 26, 33, 40–41, 49, 55 Multi-Step TAKS Prep 10, 18, 26, 33, 34, 39, 48, 54, 58 Graphic Organizers 8, 16, 24, 31, 37, College Entrance Exam Practice 65 46, 52 Homework Help Online 9, 17, 24, 31, 38, 47, 53 TAKS Tackler 66 TAKS Prep 68 Geometric Reasoning ARE YOU READY?...................................... 71 TEKS Inductive and Deductive Reasoning G.3.D 2-1 Using Inductive Reasoning to Make Conjectures........... 74 On Track for TAKS: Number Theory G.3.A G.3.E G.4.A G.3.A Venn Diagrams....................................... 80 2-2 Conditional Statements................................. 81 2-3 Using Deductive Reasoning to Verify Conjectures.......... 88 Solve Logic Puzzles................................... 94 2-4 Biconditional Statements and Definitions................. 96 MULTI-STEP TAKS PREP.................................. 102 READY TO GO ON? QUIZ..... |
............................ 103 Mathematical Proof G.3.E G.1.A G.1.A G.1.A 2-5 Algebraic Proof........................................ 104 2-6 Geometric Proof....................................... 110 Design Plans for Proofs.............................. 117 2-7 Flowchart and Paragraph Proofs........................ 118 MULTI-STEP TAKS PREP.................................. 126 READY TO GO ON? QUIZ................................. 127 EXT Introduction to Symbolic Logic.......................... 128 G.4.A Study Guide: Preview..................................... 72 Reading and Writing Math................................ 73 Study Guide: Review.................................... 130 Chapter Test........................................... 134 Problem Sol |
ving on Location............................. 140 Tools for Success KEYWORD: MG7 TOC Reading Math 73 Writing Math 78, 81, 86, 92, 96, 100, 109, 111, 115, 125 Vocabulary 71, 72, 77, 84, 91, 99, 107, 113, 122, 130 Know-It Notes 75, 76, 81, 83, 84, 89, 90, 98, 104, 106, 107, 110, 111, 112, 113, 118, 120, 122, 128 Test Prep Exercises 79, 86, 93, 101, 109, 116, 125 Multi-Step TAKS Prep 78, 85, 92, 100, 102, 109, 115, 124, 126 Graphic Organizers 76, 84, 90, 98, 107, College Entrance Exam Practice 135 113, 122 Homework Help Online 77, 84, 91, 99, 107, 113, 122 TAKS Tackler 136 TAKS Prep 138 KEYWORD: MG7 TOC Parallel and Perpendicular Lines ARE YOU READY?..................................... 143 TEKS Lines with Transversals 3-1 Lines and Angles....................................... 146 On Track for TAKS: Algebra Systems of Equations................................ 152 G.9.A G.3.C G.3.C G.2.A G.1.A G.2.A Explore Parallel Lines and Transversals......... 154 3-2 Angles Formed by Parallel Lines and Transversals......... 155 3-3 Proving Lines Parallel.................................. 162 Construct Parallel Lines............... |
............... 170 3-4 Perpendicular Lines.................................... 172 Construct Perpendicular Lines........................ 179 MULTI-STEP TAKS PREP.................................. 180 READY TO GO ON? QUIZ................................. 181 Coordinate Geometry G.7.B G.7.B G.7.B 3-5 Slopes of Lines........................................ 182 Explore Parallel and Perpendicular Lines......... 188 3-6 Lines in the Coordinate Plane........................... 190 On Track for TAKS: Data Analysis Scatter Plots and Lines of Best Fit..................... 198 MULTI-STEP TAKS PREP.................................. 200 READY TO GO ON? QUIZ................................. 201 Study Guide: Preview.................................... 144 Reading and Writing Math............................... 145 Study Guide: Review... |
................................. 202 Chapter Test........................................... 206 Tools for Success Writing Math 150, 160, 168, 177, Study Strategy 145 186, 196 Vocabulary 143, 144, 148, 175, 185, 194, 202 Know-It Notes 146, 147, 148, 155, 156, 157, 162, 163, 173, 174, 182, 184, 185, 190, 192, 193 Graphic Organizers 148, 157, 165, 174, 185, 193 Test Prep Exercises 150–151, 160–161, 168–169, 177–178, 187, 196–197 Multi-Step TAKS Prep 150, 160, 168, 176, 180, 186, 196, 200 College Entrance Exam Practice 207 TAKS Tackler 208 Homework Help Online 148, 158, 166, TAKS Prep 210 175, 185, 194 KEYWORD: MG7 TOC Triangle Congruence ARE YOU READY?..................................... 213 TEKS G.1.A G.9.B G.2.B G.10.B G.9.B G.10.B G.9.B G.10.B G.1.A G.2.B G.2.B Triangles and Congruence 4-1 Classifying Triangles................................... 216 Develop the Triangle Sum Theorem.................. 222 4-2 Angle Relationships in Triangles........................ 223 4-3 Congruent Triangles................................... 231 MULTI |
-STEP TAKS PREP.................................. 238 READY TO GO ON? QUIZ................................. 239 Proving Triangle Congruence Explore SSS and SAS Triangle Congruence........... 240 4-4 Triangle Congruence: SSS and SAS...................... 242 Predict Other Triangle Congruence Relationships.... 250 4-5 Triangle Congruence: ASA, AAS, and HL................. 252 4-6 Triangle Congruence: CPCTC............................ 260 On Track for TAKS: Algebra Quadratic Equations................................. 266 4-7 Introduction to Coordinate Proof........................ 267 4-8 Isosceles and Equilateral Triangles....................... 273 MULTI-STEP TAKS PREP.................................. 280 READY TO GO ON? QUIZ................................. 281 EXT Proving Constructions Valid............................. 282 G.2.A Study Guide: Preview............................ |
........ 214 Reading and Writing Math............................... 215 Study Guide: Review.................................... 284 Chapter Test........................................... 288 Problem Solving on Location............................. 294 Tools for Success Reading Math 215, 273 Writing Math 220, 229, 236, 248, 258, 264, 271, 278 Vocabulary 213, 214, 219, 227, 234, 245, 256, 262, 270, 276, 284 Know-It Notes 216, 217, 218, 223, 224, 225, 226, 231, 233, 242, 243, 245, 252, 254, 255, 262, 267, 269, 273, 274, 275, 276 Graphic Organizers 218, 226, 233, 245, 255, 262, 269, 276 Test Prep Exercises 221, 230, 236, 248, 258–259, 264–265, 272, 279 Multi-Step TAKS Prep 220, 229, 236, 238, 247, 258, 264, 271, 278, 280 College Entrance Exam Practice 289 TAKS Tackler 290 Homework Help Online 219, 227, 234, TAKS Prep 292 245, 256, 262, 270, 276 KEYWORD: MG7 TOC Properties and Attributes of Triangles ARE YOU READY?..................................... 297 TEKS G.3.B G.3.B G.3.B G.2.A G.7.B Segments in Triangles 5-1 Perpendicular and Angle Bisectors...................... 300 5-2 Bisectors of Triangles............ |
....................... 307 5-3 Medians and Altitudes of Triangles...................... 314 Medians and Altitudes of Triangles Special Points in Triangles...................... 321 5-4 The Triangle Midsegment Theorem...................... 322 MULTI-STEP TAKS PREP.................................. 328 READY TO GO ON? QUIZ................................. 329 Relationships in Triangles On Track for TAKS: Algebra Solving Compound Inequalities....................... 330 G.9.B G.3.B G.3.B Explore Triangle Inequalities......................... 331 5-5 Indirect Proof and Inequalities in One Triangle........... 332 5-6 Inequalities in Two Triangles............................ 340 On Track for TAKS: Algebra Simplest Radical Form................................ 346 G.8.C G.8.C G.5.D G.2.A Hands-on Proof of the Pythagorean Theorem........ 347 5-7 The Pythagorean Theorem.............................. 348 5-8 Applying Special Right Triangles...... |
.................. 356 Graph Irrational Numbers............................ 363 MULTI-STEP TAKS PREP.................................. 364 READY TO GO ON? QUIZ................................. 365 Study Guide: Preview.................................... 298 Reading and Writing Math............................... 299 Study Guide: Review.................................... 366 Chapter Test........................................... 370 Tools for Success Reading Math 299, 300 Writing Math 306, 313, 318, 325, 338, 344, 354, 361 Vocabulary 297, 298, 304, 311, 317, 324, 336, 352, 366 Know-It Notes 300, 301, 303, 307, 309, 310, 314, 317, 323, 324, 333, 334, 335, 340, 342, 350, 351, 352, 356, 358, 359 Test Prep Exercises 306, 313, 319, 326, 339, 345, 355, 362 Multi-Step TAKS Prep 305, 312, 319, 326, 328, 338, 344, 354, 361, 364 Graphic Organizers 303, 310, 317, 324, College Entrance Exam Practice 371 335, 342, 352, 359 Homework Help Online 304, 311, 317, 324, 336, 343, 352, 360 TAKS Tackler 372 TAKS Prep 374 Polygons and Quadrilaterals ARE YOU READY?... |
.................................. 377 TEKS G.2.A G.5.B Polygons and Parallelograms Construct Regular Polygons.......................... 380 6-1 Properties and Attributes of Polygons................... 382 On Track for TAKS: Algebra Relations and Functions.............................. 389 G.9.B G.3.B G.3.B Explore Properties of Parallelograms................ 390 6-2 Properties of Parallelograms............................ 391 6-3 Conditions for Parallelograms........................... 398 MULTI-STEP TAKS PREP.................................. 406 READY TO GO ON? QUIZ................................. 407 KEYWORD: MG7 TOC G.3.B G.2.A G.3.B G.2.A G.3.B Other Special Quadrilaterals 6-4 Properties of Special Parallelograms..................... 408 Predict Conditions for Special Parallelograms... 416 6-5 Conditions for Special Parallelograms................... 418 Conditions for Special Parallelograms Explore Isosceles Trapezoids................... |
. 426 6-6 Properties of Kites and Trapezoids...................... 427 MULTI-STEP TAKS PREP.................................. 436 READY TO GO ON? QUIZ................................. 437 Study Guide: Preview.................................... 378 Reading and Writing Math............................... 379 Study Guide: Review.................................... 438 Chapter Test........................................... 442 Problem Solving on Location............................. 448 Tools for Success Writing Math 379, 388, 397, 404, 414, 424, 434 Vocabulary 377, 378, 386, 395, 412, 432, 438 Know-It Notes 383, 384, 385, 391, 392, 394, 398, 399, 401, 408, 409, 411, 418, 419, 421, 427, 429, 431 Graphic Organizers 385, 394, 401, 411, Test Prep Exercises 388, 397, 405, 414–415, 425, 434–435 Multi-Step TAKS Prep 387, 396, 404, 406, 414, 424, 434, 436 College Entrance Exam Practice 443 421, 431 TAKS Tackler 444 Homework Help Online 386, 395, 402, TAKS Prep 446 412, 422, 432 Similarity KEYWORD: MG7 TOC ARE YOU |
READY?..................................... 451 TEKS Similarity Relationships G.11.B 7-1 Ratio and Proportion................................... 454 G.5.B G.5.B G.11.A G.11.B Explore the Golden Ratio....................... 460 7-2 Ratios in Similar Polygons.............................. 462 Predict Triangle Similarity Relationships........ 468 7-3 Triangle Similarity: AA, SSS, and SAS.................... 470 MULTI-STEP TAKS PREP.................................. 478 READY TO GO ON? QUIZ................................. 479 Applying Similarity Investigate Angle Bisectors of a Triangle........ 480 7-4 Applying Properties of Similar Triangles................. 481 7-5 Using Proportional Relationships........................ 488 7-6 Dilations and Similarity in the Coordinate Plane.......... 495 G.5.B G.11.B G.11.D G.11.A On Track for TAKS: Algebra Direct Variation..................................... 501 MULTI |
-STEP TAKS PREP.................................. 502 READY TO GO ON? QUIZ................................. 503 Study Guide: Preview.................................... 452 Reading and Writing Math............................... 453 Study Guide: Review.................................... 504 Chapter Test........................................... 508 Tools for Success Reading Math 453, 455, 456 Writing Math 459, 463, 466, 476, 486, 493, 499 Know-It Notes 455, 457, 462, 464, 470, 471, 473, 481, 482, 483, 484, 490, 497 Graphic Organizers 457, 464, 473, 484, Vocabulary 451, 452, 457, 465, 491, 498, 490, 497 504 Homework Help Online 457, 465, 474, 484, 491, 498 Test Prep Exercises 459, 467, 477, 487, 493, 500 Multi-Step TAKS Prep 458, 466, 476, 478, 486, 492, 499, 502 College Entrance Exam Practice 509 TAKS Tackler 510 TAKS Prep 512 KEYWORD: MG7 TOC Right Triangles and Trigonometry ARE YOU READY?..................................... 515 TEKS Trigonometric Ratios |
G.11.C 8-1 Similarity in Right Triangles............................. 518 G.2.A G.11.C Explore Trigonometric Ratios................... 524 8-2 Trigonometric Ratios................................... 525 On Track for TAKS: Algebra G.11.C Inverse Functions.................................... 533 8-3 Solving Right Triangles................................. 534 MULTI-STEP TAKS PREP.................................. 542 READY TO GO ON? QUIZ................................. 543 Applying Trigonometric Ratios G.11.C G.11.C G.11.C G.7.A 8-4 Angles of Elevation and Depression..................... 544 Indirect Measurement Using Trigonometry.......... 550 8-5 Law of Sines and Law of Cosines....................... 551 8-6 Vectors................................................ 559 MULTI-STEP TAKS PREP............... |
................... 568 READY TO GO ON? QUIZ................................. 569 EXT Trigonometry and the Unit Circle........................ 570 G.11.A Study Guide: Preview.................................... 516 Reading and Writing Math............................... 517 Study Guide: Review.................................... 572 Chapter Test........................................... 576 Problem Solving on Location............................. 582 Tools for Success Reading Math 517, 534, 570 Writing Math 523, 525, 531, 540, 548, 557, 566, 571 Vocabulary 515, 516, 521, 529, 547, 563, 572 Know-It Notes 518, 519, 520, 525, 528, 537, 546, 552, 553, 554, 561, 563 Graphic Organizers 520, 528, 537, 546, 554, 563 Test Prep Exercises 523, 532, 540, 549, 558, 567 Multi-Step TAKS Prep 522, 530, 539, 542, 548, 557, 565, 568 College Entrance Exam Practice 577 Homework Help Online 521, 529, 537, 547, 555, 563 TAKS Tackler 578 TAKS Prep 580 KEYWORD |
: MG7 TOC Extending Perimeter, Circumference, and Area ARE YOU READY?..................................... 585 TEKS G.5.A G.5.A G.8.A G.8.A G.8.A Developing Geometric Formulas On Track for TAKS: Algebra Literal Equations.................................... 588 9-1 Developing Formulas for Triangles and Quadrilaterals.... 589 Develop π............................................ 598 9-2 Developing Formulas for Circles and Regular Polygons.... 600 9-3 Composite Figures..................................... 606 Develop Pick’s Theorem for Area of Lattice Polygons..................................... 613 MULTI-STEP TAKS PREP.................................. 614 READY TO GO ON? QUIZ................................. 615 Applying Geometric Formulas G.7.A G.11.D 9-4 Perimeter and Area in the Coordinate Plane.............. 616 9-5 Effects of Changing Dimensions Proportionally.......... 622 On Track for TAKS: Probability G.8.A G.8.A Probability. |
......................................... 628 9-6 Geometric Probability.................................. 630 Use Geometric Probability to Estimate π............. 637 MULTI-STEP TAKS PREP.................................. 638 READY TO GO ON? QUIZ................................. 639 Study Guide: Preview.................................... 586 Reading and Writing Math............................... 587 Study Guide: Review.................................... 640 Chapter Test........................................... 644 Tools for Success Writing Math 596, 605, 611, 620, Study Strategy 587 626, 635 Vocabulary 585, 586, 603, 609, 633, 640 Know-It Notes 589, 590, 591, 593, 600, 601, 602, 608, 619, 623, 624, 630, 633 Graphic Organizers 593, 602, 608, 619, 624, 633 Test Prep Exercises 596–597, 605, 611–612, 621, 627, 636 Multi-Step TAKS Prep 595, 604, 610, 614, |
620, 626, 635, 638 College Entrance Exam Practice 645 TAKS Tackler 646 Homework Help Online 593, 603, 609, TAKS Prep 648 619, 625, 633 KEYWORD: MG7 TOC Spatial Reasoning ARE YOU READY?..................................... 651 TEKS G.6.A G.9.D G.6.B G.7.C Three-Dimensional Figures 10-1 Solid Geometry........................................ 654 10-2 Representations of Three-Dimensional Figures........... 661 Use Nets to Create Polyhedrons...................... 669 10-3 Formulas in Three Dimensions.......................... 670 MULTI-STEP TAKS PREP.................................. 678 READY TO GO ON? QUIZ................................. 679 Surface Area and Volume G.8.D G.9.D G.8.D G.8.D G.8.D 10-4 Surface Area of Prisms and Cylinders.................... 680 Model Right and Oblique Cylinders.................. 688 10-5 Surface Area of Pyramids and Cones.................... 689 10-6 Volume of Prisms and Cylinders............. |
............ 697 10-7 Volume of Pyramids and Cones......................... 705 On Track for TAKS: Algebra G.8.D G.11.D Functional Relationships in Formulas.................. 713 10-8 Spheres............................................... 714 Compare Surface Areas and Volumes........... 722 MULTI-STEP TAKS PREP.................................. 724 READY TO GO ON? QUIZ................................. 725 EXT Spherical Geometry.................................... 726 G.1.C Study Guide: Preview.................................... 652 Reading and Writing Math............................... 653 Study Guide: Review.................................... 730 Chapter Test........................................... 734 Problem Solving on Location............................. 740 |
Tools for Success Writing Math 653, 659, 667, 676, 686, 695, 703, 711, 720 Vocabulary 651, 657, 665, 674, 684, 693, 701, 709, 718, 730 Know-It Notes 654, 656, 664, 670, 671, 672, 673, 680, 681, 683, 689, 690, 692, 697, 699, 700, 705, 707, 708, 714, 716, 717, 726, 727 Graphic Organizers 656, 664, 673, 683, 692, 700, 708, 717 Test Prep Exercises 659, 667, 677, 687, 695, 703–704, 712, 721 Multi-Step TAKS Prep 658, 666, 675, 678, 686, 695, 703, 711, 720, 724 College Entrance Exam Practice 735 TAKS Tackler 736 Homework Help Online 657, 665, 674, TAKS Prep 738 684, 693, 701, 709, 718 KEYWORD: MG7 TOC Circles ARE YOU READY?..................................... 743 TEKS Lines and Arcs in Circles G.9.C 11-1 Lines That Intersect Circles.............................. 746 On Track for TAKS: Data Analysis Circle Graphs........................................ 755 11-2 Arcs and Chords....................................... 756 11-3 Sector Area and Arc Length....................... |
...... 764 MULTI-STEP TAKS PREP.................................. 770 READY TO GO ON? QUIZ................................. 771 Angles and Segments in Circles Inscribed Angles....................................... 772 11-4 Explore Angle Relationships in Circles.......... 780 11-5 Angle Relationships in Circles........................... 782 Explore Segment Relationships in Circles....... 790 11-6 Segment Relationships in Circles........................ 792 11-7 Circles in the Coordinate Plane.......................... 799 MULTI-STEP TAKS PREP.................................. 806 G.1.A G.8.B G.5.B G.2.A G.5.B G.9.C G.5.A G.2.B READY TO GO ON? QUIZ................................ 807 EXT Polar Coordinates...................................... 808 G.1.A Study Guide: Preview.............................. |
...... 744 Reading and Writing Math............................... 745 Study Guide: Review.................................... 810 Chapter Test........................................... 814 Tools for Success Reading Math 745, 748 Writing Math 754, 756, 762, 769, 778, 788, 797, 804 Vocabulary 743, 744, 751, 760, 767, 776, 810 Know-It Notes 746, 747, 748, 749, 750, 756, 757, 759, 764, 765, 766, 772, 773, 774, 775, 782, 783, 784, 785, 786, 792, 793, 794, 795, 799, 801 Graphic Organizers 750, 759, 766, 775, 786, 795, 801 Test Prep Exercises 754, 763, 769, 778, 789, 798, 804 Multi-Step TAKS Prep 753, 762, 768, 770, 777, 788, 797, 803, 806 College Entrance Exam Practice 815 TAKS Tackler 816 Homework Help Online 751, 760, 767, TAKS Prep 818 776, 786, 795, 802 KEYWORD: MG7 TOC Extending Transformational Geometry ARE YOU READY?..................................... 821 TEKS G.10.A G.10.A G.10.A G.10.A G.10.A Congruence Transformations 12-1 Reflections........................... |
................. 824 12-2 Translations........................................... 831 On Track for TAKS: Algebra Transformations of Functions......................... 838 12-3 Rotations............................................. 839 Explore Transformations with Matrices............ 846 12-4 Compositions of Transformations....................... 848 MULTI-STEP TAKS PREP.................................. 854 READY TO GO ON? QUIZ................................. 855 Patterns G.10.A G.5.C G.5.C G.11.A 12-5 Symmetry............................................ 856 12-6 Tessellations........................................... 863 Use Transformations to Extend Tessellations......... 870 12-7 Dilations.............................................. 872 MULTI-STEP TAKS PREP. |
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