text stringlengths 235 3.08k |
|---|
to use high-speed computers to create the figure below, called the Mandelbrot set. Only the black area is part of the set itself. The rainbow colors represent properties of points near the Mandelbrot set. To learn more about different kinds of fractals, visit www.keymath.com/DG. EXPLORATION Patterns in Fractals 137 ● ... |
–7, find the next two terms in the sequence. 4. 7, 21, 35, 49, 63, 77,?,? 6. 7, 2, 5, 3, 8, 11,?,? 5. Z, 1, Y, 2, X, 4, W, 8,?,? 7. A, 4, D, 9, H, 16, M, 25,?,? For Exercises 8 and 9, generate the first six terms in the sequence for each function rule. 8. f(n) n2 1 9. f(n) 2n1 For Exercises 10 and 11, draw the next sha... |
If 28 lines are drawn on a plane, what is the maximum number of points of intersection possible? 23. If a whole bunch of lines (no two parallel, no three concurrent) intersect in a plane 2926 times, how many lines are a whole bunch? CHAPTER 2 REVIEW 139 EW ● CHAPTER 2 REVIEW ● CHAPTER 2 REVIEW ● CHAPTER 2 REVIEW ● CHA... |
to add to your portfolio. These could include an investigation, a project, or a complex homework exercise. Make sure your work is complete. Describe why you chose the piece and what you learned from it. 140 CHAPTER 2 Reasoning in Geometry CHAPTER 3 Using Tools of Geometry There is indeed great satisfaction in acquirin... |
, angles, surfaces, and solids. He also explained why the constructions were correct with deductive reasoning. A page from a book on Euclid, above, shows some of his constructions and a translation of his explanations from Greek into Latin. When you sketch an equilateral triangle, you may make a freehand sketch of a tr... |
is a measuring tool, not a construction tool. You will need ● a compass ● a straightedge Investigation 2 Copying an Angle D E F Step 1 The first two stages for copying DEF are shown below. Describe each stage of the process. D D E F G Stage 1 E F G Stage 2 Step 2 Step 3 What will be the final stage of the construction... |
ises 7 and 8 using constructions with patty paper. 10. Draw quadrilateral QUAD. Duplicate it, using your compass and straightedge. Label the construction COPY so that QUAD COPY. 11. Technology Use geometry software to construct an equilateral triangle. Drag each vertex to make sure it remains equilateral. LESSON 3.1 Du... |
the segment. This line is its perpendicular bisector. Line is the perpendicular bisector of AB. A O B Investigation 1 Finding the Right Bisector In this investigation you will discover how to construct the perpendicular bisector of a segment. Q P Q Q P P Step 1 Step 2 Step 3 Step 1 Step 2 Step 3 Draw a segment on patt... |
segment. Use your compass to find another such point. Use these points to construct a line. Is this line the perpendicular bisector of the segment? Use the paper-folding technique of Investigation 1 to check. 148 CHAPTER 3 Using Tools of Geometry Step 4 Complete the conjecture below, and write a summary of what you di... |
side. What do you notice about the three bisectors? 8. Construct ABC. Construct medians AM, BN, and CL. Notice anything special? 9. Construct DEF. Construct midsegment GH where G is the midpoint of DF and H is the midpoint of DE. What do you notice about the relationship between EF and GH? 10. Copy rectangle DSOE onto... |
. 50° 8 65° 8 65° D. 6 103° 8 11 B. E. 4 6 4 6 115° C. 3 5 4 F. 7 59° 73° 9 8 48° 21. Sketch and label a polygon that has exactly three sides of equal length and exactly two angles of equal measure. 22. Sketch two triangles. Each should have one side measuring 5 cm and one side measuring 9 cm, but they should not be co... |
How is PA related to PB? What does this answer tell you about where point P lies? Hint: See the Converse of the Perpendicular Bisector Conjecture. Construct the perpendicular bisector of AB. Label the midpoint M. Step 1 Step 2 Step 3 Step 4 152 CHAPTER 3 Using Tools of Geometry Step 5 You have now constructed a perpen... |
is a perpendicular segment from a vertex to the opposite side or to a line containing the opposite side. Altitude Altitude Altitudes An altitude can be inside the triangle. An altitude can be outside the triangle. An altitude can be one of the sides of the triangle. The length of the altitude is the height of the tria... |
a very large acute triangle on your patty paper. Place a point inside the triangle. Now construct perpendiculars from the point to all three sides of the triangle by folding. Mark your figure. How can you use your construction to decide which side of the triangle your point is closest to? 8. Construct an isosceles rig... |
. Most Islamic designs of this kind can be constructed using only a compass and a straightedge. Try it. Use your compass and straightedge to re-create this design or to create a design of your own based on an 8-pointed star. Here are two diagrams to get you started. 156 CHAPTER 3 Using Tools of Geometry L E S S O N 3.4... |
find a method for bisecting an angle using a compass and straightedge. Each person in your group should investigate a different angle. You will need ● a compass ● a straightedge Step 1 Step 2 Step 3 Draw an angle. Find a method for constructing the bisector of the angle. Experiment! Hint: Start by drawing an arc cente... |
it is true? 13. In this lesson you discovered the Angle Bisector Conjecture. Write the converse of the Angle Bisector Conjecture. Do you think it’s true? Why or why not? Notice how this mosaic floor at Church of Pomposa in Italy (ca. 850 C.E.) uses many duplicated shapes. What constructions do you see in the square pa... |
, if they were extended, they would intersect. Therefore, they are not parallel. The lines in the third pair appear to be parallel, but if you extend them far enough in both directions, can you be sure they won’t meet? There are many ways to be sure that the lines are parallel. You will need ● patty paper Investigation... |
are many solutions!) T A R P 6. Using patty paper and straightedge, or a compass and straightedge, construct parallelogram GRAM with RG and RA as two consecutive sides and ML as the distance between RG and AM. (How many solutions can you find?) G R M R A L 162 CHAPTER 3 Using Tools of Geometry 7. Mini-Investigation Co... |
. IMPROVING YOUR VISUAL THINKING SKILLS Visual Analogies Which of the designs at right complete the statements at left? Explain. 1. is to as is to? 2. is to as is to? 3. is to as is to? A. C. A. C. A. C. B. D. B. D. B. D. 164 CHAPTER 3 Using Tools of Geometry ALGEBRA SKILLS 1 ● USING YOUR ALGEBRA SKILLS 1 ● USING YOUR ... |
the slope of each line. ) 6 ( 2 slope of AB 3 ( ) 5 1, are negative reciprocals of each other, so AB CD. and 3 The slopes, 2 2 3 ( 2) 3 slope of CD 10 4 4 2 8 6 EXAMPLE B Given points E(3, 0), F(5, 4), and Q(4, 2), find the coordinates of a point P such that PQ is parallel to EF. y Solution We know that if PQ EF, then... |
P such that PQ is parallel to AB. 6. Given C(2, 1), D(5, 4), and Q(4, 2), find two possible locations for a point P such that PQ is perpendicular to CD. For Exercises 7–9, find the slope of each side, and then determine whether each figure is a trapezoid, a parallelogram, a rectangle, or just an ordinary quadrilateral... |
— to construct a triangle? Let’s first consider a case in which only three segments are given. EXAMPLE A Construct ABC using the three segments AB, BC, and CA shown below. How many different-size triangles can be drawn? A B C B C A Solution You can begin by copying one segment, for example AC. Then adjust your compass ... |
. Given: O Construct: DOT 3. Given: Y Construct: IGY LESSON 3.6 Construction Problems 169 4. Given the triangle shown at right, construct another triangle with angles congruent to the given angles but with sides not congruent to the given sides. Is there more than one triangle with the same three angles? 5. The two seg... |
pile, and say “C.” 3. Take the third card, place it at the bottom, and say “E.” 4. You’ve just spelled ace. Now take the fourth card and turn it faceup on the table. The card should be an ace. 5. Continue in this fashion, saying “T,” “W,” and “O” for the next three cards. Then turn the next card faceup. It should be a... |
face. Draw the hidden back vertical and horizontal edges with dashed lines. Erase unnecessary lines and dashed segments. Repeat Steps 1–4 several more times, each time placing the first box face in a different position with respect to h and V—above, below, or overlapping h; to the left or right of V or centered on V. ... |
in your group. Are any faces of the box parallel to the picture plane? Does each box face have a pair of parallel sides? Explain how the viewing position is affected by the distance between V1 and V2 relative to the size of the box. Must the box be between V1 and V2? Perspective helps in designing the lettering painte... |
a triangle?. C-11 Step 5 For what kind of triangle will the points of concurrency be the same point? The point of concurrency for the three angle bisectors is the incenter. The point of concurrency for the perpendicular bisectors is the circumcenter. The point of concurrency for the three altitudes is called the ortho... |
helps to understand why it is true. Y L 1 A P 2 You can use your compass to construct a circle that passes through the three vertices of the triangle with the circumcenter as the center of the circle. So, you can also think of the circumcenter as the center of a circle that passes through the three vertices of a trian... |
aid center of Mt. Thermopolis State Park needs to be at a point that is equidistant from three bike paths that intersect to form a triangle. Locate this point so that in an emergency medical personnel will be able to get to any one of the paths by the shortest route possible. Which point of concurrency is it? You will ... |
R S 6. Construction Draw a large triangle. Construct a circle inscribed in the triangle. 7. Construction Draw a triangle. Construct a circle circumscribed about the triangle. 8. Is the inscribed circle the greatest circle to fit within a given triangle? Explain. If you think not, give a counterexample. 9. Does the cir... |
? Investigate by using geometry software. Write a paragraph describing your findings. 19. Sketch the section formed when the plane slices the cube as shown. For Exercises 20–24, match each geometric construction with one of the figures below. 20. Construction of a perpendicular bisector 21. Construction of an angle bis... |
ene triangle as possible and label it CNR, as shown at right. Locate the midpoints of the three sides. Construct the medians and complete the conjecture. C Median Concurrency Conjecture The three medians of a triangle?. R N C-14 The point of concurrency of the three medians is the centroid. Step 2 Step 3 Label the thre... |
formed by one median? Step 4 Is there a single point where you can balance the triangle? 184 CHAPTER 3 Using Tools of Geometry If you have found the balancing point for the triangle, you have found its center of gravity. State your discovery as a conjecture, and add it to your conjecture list. Center of Gravity Conjec... |
5. Construction Construct an equilateral triangle, then construct angle bisectors from two vertices, medians from two vertices, and altitudes from two vertices. What can you conclude? 6. Construction On patty paper, draw a large isosceles triangle with an acute vertex angle that measures less than 40°. Copy it onto th... |
letters represents single bonds. The triple dash ( ) between letters represents a triple bond HC H C C C HC H H H Ethyne C2H2 Propyne C3H4 Butyne C4H6 Sketch the alkyne with eight carbons in the chain. What is the general rule for alkynes CnH? hydrogen atoms (H) are in the alkyne?? In other words, if there are n carbo... |
N 1.0 You will need ● patty paper Step 1 Step 2 Step 3 The Euler Line In the previous lessons you discovered the four points of concurrency: circumcenter, incenter, orthocenter, and centroid. In this activity you will discover how these points relate to a special line, the Euler line. The Euler line is named after the... |
An obtuse triangle? A right triangle? Drag the orthocenter. Describe how this affects the triangle. Hide the altitudes. Draw segments from each vertex to the orthocenter, as shown, forming three triangles within the original triangle. Now find the orthocenter of each of the three triangles. The Geometer’s Sketchpad wa... |
. 5. If a point is equidistant from the endpoints of a segment, then it must be the midpoint of the segment. 6. The set of all the points in the plane that are a given distance from a line segment is a pair of lines parallel to the given segment. 7. It is not possible for a trapezoid to have three congruent sides. 8. T... |
Riff and Raff, in the shape of a triangular prism. Which point of concurrency in the triangular base do they need to locate in order to construct the largest possible circular entrance? 26. Adventurer Dakota Davis has a map that once showed the location of a large bag of gold. Unfortunately, the part of the map that s... |
gruent. 43. An altitude of a triangle must be inside the triangle. 44. The orthocenter of a triangle is the point of intersection of the three perpendicular bisectors of the sides. 45. If two lines are parallel to the same line, then they are parallel to each other. 46. If the sum of the measure of two angles is 180°, ... |
f 130° e CHAPTER 3 REVIEW 195 EW ● CHAPTER 3 REVIEW ● CHAPTER 3 REVIEW ● CHAPTER 3 REVIEW ● CHAPTER 3 REVIEW ● CHAPTE 63. Draw a scalene triangle ABC. Use a 64. What’s wrong with this picture? straightedge and compass to construct the incenter of ABC. A 150° C 124° E F 56° B D 26° 65. What is the minimum number of reg... |
©2002 Cordon Art B. V.–Baarn–Holland. All rights reserved In this chapter you will ● learn why triangles are so useful in structures ● discover relationships between the sides and angles of triangles ● learn about the conditions that guarantee that two triangles are congruent L E S S O N 4.1 Teaching is the art of ass... |
arrangement show the sum of the angle measures? Compare results with others in your group. State a conjecture. c b a Triangle Sum Conjecture The sum of the measures of the angles in every triangle is?. C-17 LESSON 4.1 Triangle Sum Conjecture 199 Steps 1 through 5 may have convinced you that the Triangle Sum Conjecture... |
, because A and B are congruent to D and F, then C must be congruent to E. You could also use the Triangle Sum Conjecture to find that D and F both measure 50°. Since they have the same measure, they are congruent. C D 60° 70° F E EXERCISES 1. Technology Using geometry software, construct a triangle. Use the software t... |
–21, tell whether the statement is true or false. For each false statement, explain why it is false or sketch a counterexample. D E 17. If two sides in one triangle are congruent to two sides in another triangle, then the two triangles are congruent. 18. If two angles in one triangle are congruent to two angles in anot... |
honors. The front part of the museum is a large glass pyramid, divided into small triangular windows that resemble a Sierpin´ski tetrahedron, a three-dimensional Sierpin´ski triangle. The pyramid structure rests on a rectangular tower and a circular theater that looks like a performance drum. Architect I. M. Pei (b 19... |
the Converse True? Suppose a triangle has two congruent angles. Must the triangle be isosceles? A A C B Step 1 B Step 2 Step 1 Step 2 Step 3 Draw a segment and label it AB. Draw an acute angle at point A. This angle will be a base angle. (Why can’t you draw an obtuse angle as a base angle?) Copy A at point B on the sa... |
gruence from the information given. Remember to write the statement so that corresponding parts are in order. 10. GEA? E 24 cm G 36 cm A 36 cm N 24 cm C 11. JAN? N I 50° J A 40° E C In Exercises 12 and 13, use compass and straightedge, or patty paper, to construct a triangle that is not congruent to the given triangle,... |
nine digits—1 through 9—must be used, in order! You may use any combination of signs for the four basic operations (,,, ), parentheses, decimal points, exponents, factorial signs, and square root symbols, and you may place the digits next to each other to create two-digit or three-digit numbers. Example: 1 2(3 4.5) 67... |
intercept and b is the slope y mx b, where m is the slope and b is the y-intercept 210 CHAPTER 4 Discovering and Proving Triangle Properties ALGEBRA SKILLS 4 ● USING YOUR ALGEBRA SKILLS 4 ● USING YOUR ALGEBRA SKILLS 4 ● USING YO The only difference between these two forms is the order of the x term and the constant ter... |
4 2 (0, 2) –2 –4 x 4 6 In Exercises 6–8, write an equation for the line through each pair of points. 5. y (–5, 8) (8, 2) x 6. (1, 2), (3, 4) 7. (1, 2), (3, 4) 8. (1, 2), (6, 4) 9. The math club is ordering printed T-shirts to sell for a fundraiser. The T-shirt company charges $80 for the set-up fee and $4 for each pri... |
its sides? In this lesson you will discover some geometric inequalities that answer these questions. LESSON 4.3 Triangle Inequalities 213 You will need ● a compass ● a straightedge Investigation 1 What Is the Shortest Path from A to B? Each person in your group should do each construction. Compare results when you fin... |
exterior angles. If you extend one side of a triangle beyond its vertex, then you have constructed an exterior angle at that vertex. Each exterior angle of a triangle has an adjacent interior angle and a pair of remote interior angles. The remote interior angles are the two angles in the triangle that do not share a v... |
If 54 and 48 are the lengths of two sides of a triangle, what is the range of possible values for the length of the third side? 216 CHAPTER 4 Discovering and Proving Triangle Properties 12. What’s wrong with this picture? Explain. 13. What’s wrong with this picture? Explain. 11 cm 25 cm 48 cm 72° 72° 74° 74° In Exerci... |
lengths to simulate the cutting of the straw. Analyze the results and calculate the probability based on your data. How close was your prediction? Your project should include Your prediction and an explanation of how you arrived at it. Your randomly generated data. An analysis of the results and your calculated probab... |
-Angle (SSA) Angle-Angle-Angle (AAA) Two pairs of congruent angles and one pair of congruent sides (sides not between the pairs of angles) Two pairs of congruent sides and one pair of congruent angles (angles not between the pairs of sides) Three pairs of congruent angles LESSON 4.4 Are There Congruence Shortcuts? 219 ... |
to place the triangles on top of each other and see if they coincide.) Is it possible to construct different triangles from the same three parts, or will all the triangles be congruent? D Step 3 You are now ready to complete the conjecture for the SAS case. SAS Congruence Conjecture C-25 If two sides and the included ... |
9–14, name a triangle congruent to the given triangle and state the congruence conjecture. If you cannot show any triangles to be congruent from the information given, write “cannot be determined” and explain why. 9. ANT? N L T A E F 12. MAN? N Y M A B O 10. RED? B R 11. WOM? O E D U L M W T 13. SAT? O A S T 14. GIT? ... |
celes right triangle ABC has vertices with coordinates A(8, 2), B(5, 3), and C(0, 0). Find the coordinates of the orthocenter. IMPROVING YOUR REASONING SKILLS Container Problem II You have a small cylindrical measuring glass with a maximum capacity of 250 mL. All the marks have worn off except the 150 mL and 50 mL mark... |
and ASA Congruence Conjectures are true for all pairs of triangles that have those sets of corresponding parts congruent. LESSON 4.5 Are There Other Congruence Shortcuts? 225 The ASA case is closely related to another special case—the Side-Angle-Angle (SAA) case. You can investigate the SAA case with compass and strai... |
ises 17–19, 20, and 23 1. AMD RMC D A M C R 2. BOX CAR 3. GAS IOL. HOW FEW 5. FSH FSI 6. ALT INT LESSON 4.5 Are There Other Congruence Shortcuts? 227 In Exercises 7–14, name a triangle congruent to the triangle given and state the congruence conjecture. If you cannot show any triangles to be congruent from the informat... |
? K M 24. Sketch five lines in a plane that intersect in exactly five points. Now do this in a different way. 25. APPLICATION Scientists use seismograms and a method called triangulation to pinpoint the epicenter of an earthquake. a. Data recorded for one quake show that the epicenter is 480 km from Eureka, California;... |
it is given that AM BM and A B. So, by ASA, AMD BMC. Because the triangles are congruent, AD BC by CPCTC. If you use a congruence shortcut to show that two triangles are congruent, then you can use CPCTC to show that any of their corresponding parts are congruent. When you are trying to prove that triangles are congru... |
and the lengths of horizontal and vertical segments shown on the grid. Answer the question about segment or angle congruence. If your answer is yes, explain why. 10. Is FR GT? Why? 11. Is OND OCR? Why? T R O G E F 12. In Chapter 3, you used inductive reasoning to discover how to duplicate an angle using a compass and ... |
(diagonal) to a quadrilateral you create a quadrilateral that consists of two triangles, and that makes it rigid. What is the minimum number of struts needed to make a pentagon rigid? A hexagon? A dodecagon? What is the minimum number of struts needed to make other polygons rigid? Complete the table and make your conj... |
reason for everything. TRADITIONAL SAYING So far, you have written your explanations as paragraph proofs. First, we’ll look at a diagram and explain why two angles must be congruent, by writing a paragraph proof, in Example A. Then we’ll look at a different tool for writing proofs, and use that tool to write the same ... |
clearly. It helps to mark the given information on the figure. Then state what you are trying to show. Given: AR ER EC AC Show: E A Flowchart Proof C R E 1 AR ER Given 2 3 EC AC Given RC RC Same segment A 4 RCE RCA 5 E A SSS Congruence Conjecture CPCTC In a flowchart proof, the arrows show how the logical argument flo... |
ure. Given: NEW with W E NS is an angle bisector Show: NEW is an isosceles triangle Flowchart Proof 1 NS is an angle bisector Given 2 1? Definition of? WN NE??? 7 NEW is isosceles? 4 NS NS? 6. Complete the flowchart proof. What does this proof tell you about parallelograms? Given: SA NE SE NA Show: SA NE Flowchart Proo... |
What triangle congruence shortcut is Samantha using? Explain. 16. What is the probability of randomly selecting one of the shortest diagonals from all the diagonals in a regular decagon? 17. Sketch the solid shown with the red 18. Sketch the new location of rectangle BOXY and green cubes removed. after it has been rot... |
straightedge Step 1 Step 2 Step 3 Step 4 Step 5 K Construct a large isosceles triangle on a sheet of unlined paper. Label it ARK, with K the vertex angle. Construct angle bisector KD with point D on AR. Do ADK and RDK look congruent? With your compass, compare AD and RD. Is D the midpoint of AR? If D is the midpoint, ... |
les Triangle Conjecture. So, if the Equilateral Triangle Conjecture and the Equiangular Triangle Conjecture are both true then we can combine them. Complete the conjecture below and add it to your conjecture list. Equilateral/Equiangular Triangle Conjecture Every equilateral triangle is?, and, conversely, every equiang... |
B A 6 1 and 2 are right angles Congruent supplementary angles are 90° 7 CD AB? 8? Definition of altitude 244 CHAPTER 4 Discovering and Proving Triangle Properties 6. Create a flowchart proof for Conjecture C. Conjecture C: The bisector of the vertex angle in an isosceles triangle is also the median to the base. Given:... |
. Can you find any other points that would create a right triangle? 16. APPLICATION Hugo hears the sound of fireworks three seconds after he sees the flash. Duane hears the sound five seconds after he sees the flash. Hugo and Duane are 1.5 km apart. They know the flash was somewhere to the north. They also know that a ... |
triangle falls inside your triangle, undo and try again, selecting the two endpoints in reverse order. Connect the centroids of the equilateral triangles. Triangle GQL is called the outer Napoleon triangle of ABC. Drag the vertices and the sides of ABC and observe what happens. Portrait of Napoleon by the French paint... |
y-axis and whose line of symmetry is the x-axis. What is the equation of the line containing the other side? 4. Graph the lines y 2x 2, y 1 x 1, y x, and y x. Describe the figure that 2 the lines form. Find other sets of lines that form figures like this one. 248 CHAPTER 4 Discovering and Proving Triangle Properties V... |
4 REVIEW ● CHAPTER 4 REVIEW ● CHAPTE 10. TIM? 11. TRP? 12. CAT? E I M T 13. CGH? H I G C N R T A P 14. AB CD ABE? B A E D C T C R A 15. Polygon CARBON is a regular hexagon. ACN? N O C B A R 16.??, AD? 17.?? 18 19.??, TR? 20.??, EI? 22.?? Is NCTM a parallelogram or a trapezoid? 23.?? LAI is isosceles with IA LA. M T N ... |
CHAPTER 4 REVIEW ● CHAPTER 4 REVIEW ● CHAPTE For Exercises 33 and 34, use the segments and the angles below. Use either patty paper or a compass and a straightedge. The lowercase letter above each segment represents the length of the segment. x y z P A 33. Construction Construct PAL given P, A, and AL y. 34. Construct... |
Describe your findings. 5. Is there a conjecture (similar to the Triangle Exterior Angle Conjecture) that you can make about exterior and remote interior angles of a convex quadrilateral? Experiment. Write about your findings. 6. Is there a conjecture you can make about inequalities among the sums of the lengths of si... |
another’s problems. Then discuss the problems in your group: Were they representative of the content of the chapter? Were some too hard or too easy? Writing your own problems is an excellent way to assess and review what you’ve learned. Maybe you can even persuade your teacher to use one of your items on a real test! ... |
your group should draw a different version of the same polygon. For example, if your group is investigating hexagons, try to think of different ways you could draw a hexagon. Step 1 Step 2 Step 3 Draw the polygon. Carefully measure all the interior angles, then find the sum. Share your results with your group. If you ... |
sides of equiangular polygon Measures of each angle of equiangular polygon In Exercises 3–8, use your conjectures to calculate the measure of each lettered angle. 3. a? 4. b? 76° 72° a 70° b 110° 116° 68° 5. e? f? e f LESSON 5.1 Polygon Sum Conjecture 257 6. c? d? 7. g? h? 8. j? k? d c 44° 78° 30° 66° 130° 108° h 117°... |
a quadrilateral and locate the midpoints of its four sides. Construct segments connecting the midpoints of opposite sides. Construct the point of intersection of the two segments. Drag a vertex or a side so that the quadrilateral becomes concave. Observe these segments and make a conjecture. 20. Write the equation of ... |
Use the Polygon Sum Conjecture to calculate the measure of the remaining interior angle. Check your answer using your protractor. Use the Linear Pair Conjecture to calculate the measure of each exterior angle. Calculate the sum of the measures of the exterior angles. Share your results with your group members. 260 CHA... |
b 44° c b 56° a d 94° 69° 6. 9. d c 86° g 39° d f c e a k 18° h b 10. How many sides does a regular polygon have if each exterior angle measures 24°? 11. How many sides does a polygon have if the sum of its interior angle measures is 7380°? 12. Is there a maximum number of obtuse exterior angles that any polygon can h... |
ons -pointed star ABCDE 6-pointed star FGHIJK Draw five points A through E in a circular path, clockwise. Connect every second point with AC, CE, EB, BD, and DA. Measure the five angles A through E at the star points. Use the calculator to find the sum of the angle measures. Drag each vertex of the star and observe wha... |
celes triangle, the vertex angle is the angle between the two congruent sides. Therefore, let’s call the two angles between each pair of congruent sides of a kite the vertex angles of the kite. Let’s call the other pair the nonvertex angles. Nonvertex angles Vertex angles You will need ● patty paper Step 1 Step 2 A kit... |
oid is a quadrilateral with exactly one pair of parallel sides. In a trapezoid the parallel sides are called bases. A pair of angles that share a base as a common side are called base angles. Pair of base angles In the next investigation, you will discover some properties of trapezoids. LESSON 5.3 Kite and Trapezoid Pr... |
oid are congruent? Let’s continue. Step 7 Draw both diagonals. Compare their lengths. Share your observations with others near you. Step 8 Copy and complete the conjecture. Isosceles Trapezoid Diagonals Conjecture The diagonals of an isosceles trapezoid are?. C-40 C-41 Suppose you assume that the Isosceles Trapezoid Co... |
graphing calculator. This is done with parametric equations, which give the x- and y-coordinates of a point in terms of a third variable, or parameter, t. Set your calculator’s mode to degrees and parametric. Set a friendly window with an x-range of 4.7 to 4.7 and a y-range of 3.1 to 3.1. Set a t-range of 0 to 360, an... |
Draw the midsegments. You should now have four small triangles. Place a second piece of patty paper over the first and copy one of the four triangles. Compare all four triangles by sliding the copy of one small triangle over the other three triangles. Compare your results with the results of your group. Copy and compl... |
lengths of the two bases by marking a point on the extension of the longer base. Step 1 Step 2 Step 3 Step 4 Step 5 Step 6 274 CHAPTER 5 Discovering and Proving Polygon Properties Sum Step 5 Step 6 Step 7 Step 7 How many times does the midsegment fit onto the segment representing the sum of the lengths of the two base... |
midpoints? Use the Triangle Midsegment Conjecture to explain your answer. 10. Deep in a tropical rain forest, archaeologist Ertha Diggs and her assistant researchers have uncovered a square-based truncated pyramid (a square pyramid with the top part removed). The four lateral faces are isosceles trapezoids. A line of ... |
(5, 0) S (10, 0) x C (0, 0) x A (15, 0) R (?,?) LESSON 5.4 Properties of Midsegments 277 17. Find the coordinates of midpoints E and Z. Show that the slope of the line containing midsegment EZ is equal to the slope of the line containing YT. 18. Construction Use the kite properties you discovered in Lesson 5.3 to cons... |
lesson you will discover some special properties of parallelograms. A parallelogram is a quadrilateral whose opposite sides are parallel. Rhombuses, rectangles, and squares all fit this definition as well. Therefore, any properties you discover for parallelograms will also apply to these other shapes. However, to be s... |
Step 6 Step 7 Finally, let’s consider the diagonals of a parallelogram. Construct the diagonals LV and EO, as shown below. Label the point where the two diagonals intersect point M. Measure LM and VM. What can you conclude about point M? Is this conclusion also true for diagonal EO? How do the diagonals relate? E V M ... |
. What is the measure of each angle in the isosceles trapezoid face of a voussoir in this 15-stone arch? a? b? LESSON 5.5 Properties of Parallelograms 283 19. Is XYW WYZ? Explain. 20. Sketch the section formed when this pyramid is sliced by the plane. 58° W 83° 58° X Z 83° 39° 39° Y 21. Technology Construct two segment... |
12x 2y 1 Start by solving the first equation for y to get y 4x 7. Now, substitute the expression 4x 7 from the resulting equation for y in the second original equation. 12x 2y 1 12x 2(4x – 7) 1 x 3 4 Substitute 4x 7 for y. Second original equation. Solve for x. EXAMPLE A Solution To find y, substitute 3 for x in eithe... |
6 For Exercises 5 and 6 solve the systems. What happens? Graph each set of equations and use the graphs to explain your results. 5. x 6y 10 1 x 3y 5 2 6. 2x y 30 y 2x 1 7. A snowboard rental company offers two different rental plans. Plan A offers $4/hr for the rental and a $20 lift ticket. Plan B offers $7/hr for the... |
? Share your results with your group. Copy and complete the conjecture. Double-Edged Straightedge Conjecture C-49 If two parallel lines are intersected by a second pair of parallel lines that are the same distance apart as the first pair, then the parallelogram formed is a?. LESSON 5.6 Properties of Special Parallelogr... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.