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Geometry and Measurement FLORIDA CHAPTER 9 Name Class Date Lesson Student Textbook MA.8.G.2.2 9-1 Angle Relationships 345 –352 400 – 404 MA.8.G.2.2 9-2 Parallel and Perpendicular Lines 353 –360 405 – 408 MA.8.G.2.3 9-3 Triangles 361 –368 409 – 412 MA.8.G.2.3 9-4 Polygons 369 –376 413 – 417 Remember It? Copyright © by Holt McDougal. All rights reserved. Worktext 379 –380 Rev. MA.7.G.2.1 9-5 Volume of Prisms and Cylinders 420 – 424 Rev. MA.7.G.2.1 9-6 Volume of Pyramids and Cones 425 – 429 Rev. MA.7.G.2.1 9-7 Surface Area of Prisms and Cylinders 430 – 433 Rev. MA.7.G.2.1 9-8 Surface Area of Pyramids and Cones 434 – 437 MA.8.G.5.1 9-9 Scaling Three Dimensional Figures 381 –388 438 – 441 MA.8.G.5.1 9-10 Measurement in ThreeDimensional Figures 389 –396 442 – 445 Study It! 399 – 401 Write About It! 402 Chapter 9 Geometry and Measurement 343 CHAPTER Benchmark 9 Chapter at a Glance Vocabulary Connections LA.8.1.6.5 The student will relate new vocabulary to familiar words. Key Vocabulary Vocabulario Vokabilè equilateral triangle triángulo equilátero triyang ekilateral indirect measurement medición indirecta mezi endirèkt lateral surface superficie lateral sipèfisi lateral parallel lines rectas paralelas liy paralèl perpendicular lines rectas perpendiculares liy pèpandikile polygon polígono poligòn surface area área total sòm total sipèfisi transversal transversal transvèsal volume volumen volim CHAPTER To become familiar with some of the vocabulary terms in the chapter, consider the following. You may refer to the chapter, the glossary, or a dictionary if you like. 1. The word equilateral contains the prefix equi-, which means “equal,” and lateral, which means “of the side.” What do you suppose an equilateral triangle is? 3. The word lateral is from the Latin laterals, meaning “of the side” A lateral in football is a pass to the side. What do you think the lateral surface of a cone or cylinder might be? 344 Chapter 9 Geometry and Measurement Copyright © by Holt McDougal. All rights reserved. 9 2. The Greek prefix poly- means “many,” and the suffix -gon means “angle.” What do you suppose a polygon is? Explore It! Learn It! Name Summarize It! Practice It! Apply It! Class Explore It! 9-1 Date MA.8.G.2.2 Classify and determine the measure of angles,.... Angle Relationships Investigate Angle Pairs When two lines intersect, they form pairs of angles that have special relationships with one another. Activity 1 Use a protractor to measure the angles in the figures below to the nearest degree. Record the measures in the table. Figure 1 Figure 2 1 4 1 4 2 3 2 Copyright © by Holt McDougal. All rights reserved. 3 Measure of Angle 1 Measure of Angle 2 Measure of Angle 3 Measure of Angle 4 (in degrees) (in degrees) (in degrees) (in degrees) Figure 1 Figure 2 2 Add the measures of the angles and record the sums in the table. Sum of Angles 1 and 2 (in degrees) Sum of Angles 3 and 4 (in degrees) Sum of Angles 1 and 4 (in degrees) Sum of Angles 2 and 3 (in degrees) Figure 1 Figure 2 9-1 Angle Relationships 345 Explore It! Learn It! Summarize It! Practice It! Apply It! Try This 1. Compare the angle measures you recorded for Figure 1. What do you notice? 2. Compare the angle measures you recorded for Figure 2. What do you notice? Draw Conclusions 3. If two lines intersect, they form two pairs of vertical angles. Vertical angles do not share any sides. Name the two pairs of vertical angles in the figure at the right. 6 7 8 9 4. What can you say about the measures of a pair of vertical angles? For 5 –7, use the figure at the right. 5. Without using a protractor, find the measure of ∠1. Explain how you found your answer. 1 50° 7. Describe two ways you could find the measure of ∠3. 346 9-1 Angle Relationships Copyright © by Holt McDougal. All rights reserved. 6. Without using a protractor, find the measure of ∠2. Explain how you found your answer. 2 3 Explore It! Learn It! Summarize It! Name Practice It! Class Learn It! Apply It! 9-1 Date MA.8.G.2.2 Classify and determine the measure of angles,…. Angle Relationships (Student Textbook pp. 400–404) Lesson Objective Classify angles and find their measures Vocabulary angle right angle acute angle obtuse angle straight angle complementary angles Copyright © by Holt McDougal. All rights reserved. supplementary angles adjacent angles vertical angles congruent angles Example 1 Use th U the diagram di to name each figure. A. two acute angles B. two obtuse angles m∠SQP = S T ° , m∠RQT = ° 43° 90 47° ° Lesson Tutorial Videos @ thinkcentral.com P Q R 9-1 Angle Relationships 347 Learn It! Explore It! Summarize It! Practice It! C. a pair of complementary angles Apply It! S T m∠TQP + m∠RQS ° = ° + ° 43° 90 47° P = 90° Q R D. two pairs of supplementary angles ° m∠TQP + m∠TQR = ° m∠SQP + m∠SQR = Check It Out! + + ° ° = 180° = 180° A Use the diagram to name each figure. B 1a. two acute angles 1b. two obtuse angles 58° E 32° C D 1c. a pair of complementary angles 1d. two pairs of supplementary angles 1 Use th the di diagram to find each angle measure. U A. If m∠1 = 37°, find m∠2. ° ° + m∠2 = -37° −−−−−−−−−− -37° −−−−− m∠2 = 348 9-1 Angle Relationships 4 3 B. Find m∠3. ° m∠1 + m∠2 = 2 ° . ° + m∠3 = -143° −−−−−−−−−− ° ° m∠2 + m∠3 = m∠2 = . -143° −−−−− ° Lesson Tutorial Videos @ thinkcentral.com Copyright © by Holt McDougal. All rights reserved. Example 2 Explore It! Check It Out! Learn It! Summarize It! Practice It! Apply It! U the diagram to find each angle measure. Use 2a. If m∠3 = 142°, find m∠4. 2b. Find m∠1. 2 3 1 4 Example 3 A ttraffic ffi engineer i designed a section of roadway where three streets intersect. Based on the diagram, what is the measure of ∠DBE? A 26° C B F D E Step 1: Find m∠CBD. ∠CBD ∠ angles are congruent. Copyright © by Holt McDougal. All rights reserved. m∠CBD = m∠ ° m∠CBD = Step 2: Find m∠ Congruent angles have the same measure. ° Substitute for m∠ABF. . m∠CBD + m∠ = ° + m∠DBE = 90° m∠DBE = ° Substitute ° . The angles are Subtract ° ° for m∠CBD. from both sides. Check It Out! 3. Based on the map, what is the measure of ∠BGC ? 1st St. A B Lesson Tutorial Videos @ thinkcentral.com F Main St. 42° G C E D 9-1 Angle Relationships 349 Explore It! 9-1 Learn It! Summarize It! Name Practice It! Apply It! Class Date LA.8.2.2.3 The student will organize information to show understanding or relationships among facts… (e.g., representing key points…through… summarizing…) Summarize It! Angle Relationships Think and Discuss 1. Draw a pair of angles that are adjacent but not supplementary. 2. Explain why vertical angles must always be congruent. 3. Get Organized Complete the graphic organizer. Fill in the boxes by writing the definition and making a sketch of each angle relationship. Angle Relationships Supplementary Angles Adjacent Angles Vertical Angles Definition Definition Definition Definition Sketch Sketch Sketch Sketch 350 9-1 Angle Relationships Copyright © by Holt McDougal. All rights reserved. Complementary Angles Explore It! Learn It! Summarize It! Name Practice It! Apply It! Class 9-1 Date Practice It! MA.8.G.2.2 Classify and determine the measure of angles,…. Angle Relationships Use the figure at the right for Exercises 1–7. B 1. Name a right angle in the figure. C 2. Name two acute angles in the figure. A D F G 3. Name two obtuse angles in the figure. E 4. Name an angle adjacent to ∠CFD. 5. Name a pair of complementary angles in the figure. Copyright © by Holt McDougal. All rights reserved. 6. Name three pairs of supplementary angles in the figure. 7. Suppose that ∠CFD measures 49°. What is the measure of ∠GFE? Justify your answer. Use the figure at the right for Exercises 8–9. 8. If m∠1 = 24°, find m∠2. 1 2 4 3 9. Find m∠3. 9-1 Angle Relationships 351 Explore It! 9-1 Learn It! Summarize It! Name Practice It! Class Apply It! Apply It! Date MA.8.G.2.2 Classify and determine the measure of angles… Angle Relationships The figure shows a ray of light being reflected off a mirror. The angle of incidence is congruent to the angle of reflection. Use the figure for 1−5. N Angle of incidence A 1 C 6. The figure shows a drawbridge when the moving section of the bridge is fully elevated. This section of the bridge moves through 6° in 5 seconds. How long does it take this section of the bridge to swing from the horizontal position to the fully-elevated position? Angle of reflection 65° 2 3 M B 4 Mirror D 108° 1. Find m∠3. 2. Find m∠4. The figure shows the braces in a bookshelf. Use the figure for 7−8. 3. What is m∠AMD ? What type of angle is ∠AMD ? 4 3 2 4. Name two pairs of supplementary angles in the figure. 5. Keri states that ∠1 and ∠4 are adjacent angles. Do you agree or disagree? Why? 352 9-1 Angle Relationships 7. The ratio of m∠2 to m∠1 is 5:1. What are the measures of these two angles? 8. Gridded Response Angles 3 and 4 are supplementary, and m∠4 is 30° greater than m∠3. What is the measure, in degrees, of ∠4? Copyright © by Holt McDougal. All rights reserved. 1 Explore It! Learn It! Summarize It! Name Practice It! Class Apply It! 9-2 Date MA.8.G.2.2 Classify and determine the measure of angles, including angles created when parallel lines are cut by transversals. Explore It! Parallel and Perpendicular Lines Investigate Parallel and Perpendicular Lines Parallel lines are lines in a plane that never intersect. A transversal is a line that intersects two or more lines. Angles formed by parallel lines and a transversal have some special properties. Parallel lines Transversal Activity 1 1 Use a straightedge to draw a non-vertical transversal through the given lines. Eight angles will be formed. Number the angles from ∠1 through ∠8 from left to right and then down. Copyright © by Holt McDougal. All rights reserved. 2 Use a protractor to measure the eight angles. Record the measures in the table. Angle ∠1 ∠2 ∠3 ∠4 ∠5 ∠6 ∠7 ∠8 Measure 3 Describe any relationships you notice in the table. m Try This 3 Lines m and n are parallel. Write the measure of each angle. n 1. ∠1 2. ∠2 3. ∠3 4. ∠4 5. ∠5 6. ∠6 2 1 4 6 5 135° 7 7. ∠7 9-2 Parallel and Perpendicular Lines 353 Explore It! Learn It! Summarize It! Practice It! Apply It! Draw Conclusions 8. If one of the angles formed when a transversal intersected two parallel lines measures 38˚, how many other angles would you expect to measure 38˚? 9. If a transversal intersects two parallel lines and you know one angle measures 72˚, explain how you know every other angle measure. Perpendicular lines intersect at 90˚ angles. Angles formed by perpendicular lines also have unique relationships. Activity 2 1 Use a protractor to tell whether the lines at the right are perpendicular. Explain your reasoning. m Try This Lines m and n are perpendicular. Write the measure of the angle. 11. ∠2 n 40° 1 5 12. ∠3 13. ∠4 14. ∠5 Draw Conclusions 15. Describe a method you could use to draw a line perpendicular to a given line p. 16. Describe a method you could use to draw a line parallel to a given line p. 354 9-2 Parallel and Perpendicular Lines 2 4 3 Copyright © by Holt McDougal. All rights reserved. 10. ∠1 Explore It! Learn It! Summarize It! Name Practice It! Apply It! Class 9-2 Date MA.8.G.2.2 Classify and determine the measure of angles, including angles created when parallel lines are cut by transversals. Learn It! Parallel and Perpendicular Lines (Student Textbook pp. 405–408) Lesson Objective Identify parallel and perpendicular lines and the angles formed by a transversal Vocabulary parallel lines perpendicular lines transversal Example 1 M Measure the th angles formed by the transversal and the parallel lines. Which angles seem to be congruent? ∠1, ∠3, ∠5, and ∠7 all measure 150°. Copyright © by Holt McDougal. All rights reserved. ∠2, ∠4, ∠6, and ∠8 all measure a 2 1 3 4 6 °. 5 7 b 8 c Check It Out! 1. Measure the angles formed by the transversal and the parallel lines. Which angles appear to be congruent? 1 2 3 4 5 Lesson Tutorial Videos @ thinkcentral.com 6 7 8 9-2 Parallel and Perpendicular Lines 355 Learn It! Explore It! Summarize It! Practice It! Apply It! Example 2 In the figure, line l || line m. Find the measure of each angle. Justify your answer. 124° 1 2 3 4 5 6 7 l m A. ∠4 ° m∠4 = The 124° angle and ∠4 are angles, so they are . B. ∠2 ° m∠2 + 124° = - ° ∠2 is to the 124° angle. ° ° m∠2 = C. ∠6 -124° m∠6 = ° ∠6 and ∠4 are supplementary angles. -124° ° 356 9-2 Parallel and Perpendicular Lines Lesson Tutorial Videos @ thinkcentral.com Copyright © by Holt McDougal. All rights reserved. °= m∠6 + Explore It! Learn It! Summarize It! Practice It! Apply It! Check It Out! In the figure, line n || line m. Find the measure of each angle. Justify your answer. 1 3 144° 4 5 6 7 m 8 n 2a. ∠5 Copyright © by Holt McDougal. All rights reserved. 2b. ∠7 2c. ∠8 2d. ∠6 Lesson Tutorial Videos @ thinkcentral.com 9-2 Parallel and Perpendicular Lines 357 Explore It! 9-2 Learn It! Name Summarize It! Practice It! Class Summarize It! Apply It! Date LA.8.2.2.3 The student will organize information to show understanding or relationships among facts… (e.g., representing key points…through… summarizing…) Parallel and Perpendicular Lines Think and Discuss 1. Tell how many different angles would be formed by a transversal intersecting three parallel lines. How many different angle measures would there be? 2. Explain how a transversal could intersect two other lines so that corresponding angles are not congruent. 3. Get Organized Complete the graphic organizer. Lines m and n are parallel. To fill in each box, tell whether the angle is congruent to ∠1 or ∠2 and explain why. m ∠2 358 9-2 Parallel and Perpendicular Lines Copyright © by Holt McDougal. All rights reserved. ∠1 n Explore It! Learn It! Practice It! Summarize It! Name Apply It! Class 9-2 Date MA.8.G.2.2 Classify and determine the measure of angles, including angles created when parallel lines are cut by transversals. Practice It! Parallel and Perpendicular Lines In the diagram, lines w and z are parallel. Use the diagram to answer each question. Explain your reasoning. 1. Measure the angles in the figure. Which angles appear to be congruent? 1 5 2 w 6 7 3 8 4 z 2. If m∠1 = 45°, what is m∠3? 3. If m∠2 = 132°, what is m∠7? 4. If m∠4 = 118°, what is m∠5? Copyright © by Holt McDougal. All rights reserved. 5. If m∠6 = 53°, what is m∠7? 6. If m∠8 = 28°, complete two different ways of finding m∠2. Method 1: ∠8 and ∠6 and m∠6 = angles, so they are . ∠6 and ∠2 are + . = 180°, so m∠2 = 180° - . angles, so m∠8 + m∠4 = Method 2: ∠8 and ∠4 and m∠4 = 180° - = = 152°. and ∠2 are corresponding angles, so m∠2 = = . 9-2 Parallel and Perpendicular Lines 359 Explore It! 9-2 Learn It! Name Summarize It! Apply It! Practice It! Class Date MA.8.G.2.2 Classify and determine the measure of angles, including angles created when parallel lines are cut by transversals. Apply It! Parallel and Perpendicular Lines The figure shows painted lines marking parallel parking spaces in a parking lot. Use the figure for 1−5. The figure shows several city streets. Grove Street is parallel to Hayes Street. Use the figure for 6−8. Laguna St. 1 2 3 4 12 11 10 9 5 6 8 7 123° 1. Name all the angles that are corresponding angles to ∠4. 2 1 Octavia St. Grove St. 3 Hayes St. Belden Lane 6. Suppose m∠2 is twice m∠1. Find m∠1. 2. What type of angles are ∠5 and ∠9? What can you conclude about their measures? 4. Name all angles congruent to ∠1. 5. m∠1 = 118˚. Explain how to find m∠10. 360 9-2 Parallel and Perpendicular Lines 8. Extended Response Daryl notices that Belden Lane is a transversal that intersects Laguna Street and Octavia Street. He sees that ∠2 and ∠3 are alternate interior angles and he concludes that these angles must be congruent. Do you agree or disagree with his conclusion. Why? Copyright © by Holt McDougal. All rights reserved. 3. What type of angles are ∠3 and ∠7? What can you conclude about their measures? 7. Angles 1 and 3 are complementary. Find m∠3. Explore It! Learn It! Name Summarize It! Practice It! 9-3 Apply It! Class Date MA.8.G.2.3 Demonstrate that the sum of the angles in a triangle is 180 degrees and apply this fact to find unknown measure of angles… Explore It! Triangles Investigate the Angles of a Triangle In the following activities, you’ll see three different ways that the sum of the measures of the angles of a triangle is 180°. Activity B 1 Use a protractor to measure the angles of ABC. m∠A = m∠B = m∠C = A C 2 What is the sum of the measures of the angles of ABC? 3 Draw a triangle on a sheet of paper. Make each side at least 3 inches long. Label the angles 1, 2, and 3, as shown in the triangle at the right. Copyright © by Holt McDougal. All rights reserved. 4 Cut out your triangle. Then tear off corners containing ∠1 and ∠3. 5 Arrange the corners as shown. Place ∠1 adjacent to ∠2. Place ∠3 on the other side of ∠2 so that it is also adjacent to ∠2. 2 1 3 1 2 3 6 What type of angle do the rearranged angles appear to form? 7 Use your results from Steps 2 and 6 to write a rule that describes the sum of the angle measures in a triangle. 9-3 Triangles 361 Explore It! Learn It! Summarize It! Practice It! Apply It! Try This Tell whether the three angle measures could be the angle measures of a triangle. 1. 5˚, 80˚, 90˚ 2. 30˚, 70˚, 80˚ 3. 40˚, 54˚, 86˚ 4. 37˚, 42˚, 102˚ Activity 2 1 Choose three angle measures that you think could form a triangle. Use a protractor to draw the triangle on a separate sheet of paper. Again, make each side at least 3 inches long. Write the angle measures that you chose: m∠1 = m∠2 = m∠3 = 2 Place a point in the middle of two sides of the triangle. Do this by measuring the lengths of the two sides with a ruler and dividing by 2. Draw a line connecting the two points. 3 Fold the triangle’s top vertex (A) down across the line you have drawn so that the vertex touches the bottom side of the triangle. Then fold the other two vertices (B and C) inward to meet A in its new position. 4 Explain how your results confirm the sum of the measures of the angles of your triangle. A B B 5. One angle in a right triangle measures 25°. What are the measures of the other two angles? Draw Conclusions 6. How can you find the measures of each angle of an equilateral triangle? Explain. 7. If you know the measures of two angles of a triangle, how can you find the measure of the third angle? 362 9-3 Triangles A C Copyright © by Holt McDougal. All rights reserved. Try This C Learn It! Explore It! Summarize It! Name Practice It! Class Apply It! 9-3 Date MA.8.G.2.3 Demonstrate that the sum of the angles in a triangle is 180 degrees and apply this fact to find unknown measure of angles…. Learn It! Triangles (Student Textbook pp. 409–412) Lesson Objective Find unknown angle measures in triangles Vocabulary Triangle Sum Theorem acute triangle right triangle obtuse triangle equilateral triangle Copyright © by Holt McDougal. All rights reserved. isosceles triangle scalene triangle Example 1 A. Find c° in the right triangle. B. Find m° in the obtuse triangle. c 62° 42° °+ m° 23° ° + c° = ° °+ ° + c° = ° ° + m° = - 132° - 132° c° = ° + m° = - 85° ° Lesson Tutorial Videos @ thinkcentral.com ° ° - 85° m° = ° 9-3 Triangles 363 Learn It! Explore It! Summarize It! Practice It! Apply It! Check It Out! 1a. Find b° in the right triangle. 1b. Find x ° in the right triangle. 25° 36° 38° x b Example 2 B. Find the angle measures in the scalene triangle. t° 2x° t° 62° 3x° 2t° + 62° = ° 2t° = 118° ° 10x° = _____ 180° _____ ° The angle measures are 59°, 59°, and 62° 364 9-3 Triangles 3x° + 5x° + 2x° = 10x° = 180° 2t° = _____ 118° _____ t° = 5x° x° = ° The angle measures are 3(18)° = 54°, 5(18)° = 90°, and 2(18)° = 36°. Lesson Tutorial Videos @ thinkcentral.com Copyright © by Holt McDougal. All rights reserved. A. Find A Fi d the h angle measures in the isosceles triangle. Explore It! Learn It! Summarize It! Practice It! Apply It! Check It Out! 2a. Find the angle measures in the isosceles triangle. 2b. Find the angle measures in the scalene triangle. p° + 8° 8p° 34p° 36° m° m° Example 3 Th second The d angle in a triangle is six times as large as the first. The third angle is half as large as the second. Find the angle measures and draw a possible picture. Let x ° = the first angle measure. Then 6x ° = second angle measure, and __12 (6x °) = 3x ° = third angle measure. Copyright © by Holt McDougal. All rights reserved. x ° + 6x ° + 3x ° = ° Triangle 10x ° 180° _____ = _____ x° = The angles measure Theorem Simplify, then divide both sides by . ° 54° °, °, and °. 18° 108° Check It Out! 3. The second angle in a triangle is twice as large as the first. The third angle measure is the average of the first two angle measures. Find the angle measures and draw a possible figure. Lesson Tutorial Videos @ thinkcentral.com 9-3 Triangles 365 Explore It! 9-3 Summarize It! Learn It! Name Practice It! Class Summarize It! Apply It! Date LA.8.2.2.3 The student will organize information to show understanding or relationships among facts… (e.g., representing key points… through… summarizing…) Triangles Think and Discuss 1. Explain whether a right triangle can be equilateral. Can it be isosceles? scalene? 2. Explain whether a triangle can have 2 right angles. Can it have 2 obtuse angles? 3. Get Organized Complete the graphic organizer with three kinds of triangles. Triangle Sum Theorem Example 366 9-3 Triangles Example Example Copyright © by Holt McDougal. All rights reserved. Definition Explore It! Learn It! Practice It! Summarize It! Name Class Apply It! 9-3 Date MA.8.G.2.3 Demonstrate that the sum of the angles in a triangle is 180degrees and apply this fact to find unknown measure of angles… Practice It! Triangles Find the missing angle in the triangle and tell whether the triangle is acute, obtuse, or right. 1. 56° 2. 50° 3. x° 27° 47° x° 44° 46° x° 4. 5. 6. x° 55° 35° 15° x° 137° x° x° Copyright © by Holt McDougal. All rights reserved. x° 7. The angle measures of a triangle are x°, 2x°, and 12x°. Find each angle measure. 8. The angle measures of a triangle are 2x + 10°, 3x°, and 3x + 10°. Find each angle measure. 9. The second angle in a triangle is two thirds as large as the first. The third angle is one half as large as the second angle. Find the angle measures. Draw a possible picture of the triangle. 9-3 Triangles 367 Explore It! 9-3 Learn It! Name Summarize It! Apply It! Practice It! Class Date MA.8.G.2.3 Demonstrate that the sum of the angles in a triangle is 180degrees and apply this fact to find unknown measure of angles… Apply It! Triangles The figure shows the sails of a sailboat. Use the figure for 1−2. 68° 1 6. The sum of two side lengths of a triangle must be longer than the third side. Elliott states that the distance from Jacksonville to Tampa is 210 miles. Do you think Elliott’s statement is true? Why or why not? 132° Jacksonville 2 125 mi 1. Find m∠1. 2. Find m∠2. Orlando 77 mi Tampa 3. While making a quilt, Larissa cuts a triangle out of a larger piece of material. Two of the triangle’s angles measure 51° and 36°. Explain how you can classify this triangle. 62° m 80° 1 4. A pennant has the shape of an isosceles triangle. The two congruent angles both have measures that are twice as large as the remaining angle. What are the angle measures of the pennant? 5. Corey makes an earring by bending a piece of wire into a triangle. The second angle is 3 times as large as the first. The third angle is 2 times as large as the second. What is the measure of the largest angle? 368 9-3 Triangles The beams formed by lines m and n are parallel. What is the measure, in degrees, of ∠1? n Copyright © by Holt McDougal. All rights reserved. 7. Gridded Response The figure shows part of the support structure of a bridge. Explore It! Learn It! Summarize It! Name Practice It! Apply It! Class 9-4 Date MA.8.G.2.3 Demonstrate that the sum of the angles in a triangle is 180 degrees and apply this fact to find unknown measure of angles, and the sum of angles in polygons. Explore It! Polygons Investigate Angles of Polygons You found that the measures of the angles of a triangle have a sum of 180°. In this activity, you’ll explore whether the angles of a quadrilateral also have a constant sum. Activity 1 Use a protractor. B 1 Measure the angles of quadrilateral ABCD. m∠A m∠B m∠C m∠D C A D Find the sum of the measures of the angles of quadrilateral ABCD. F Copyright © by Holt McDougal. All rights reserved. 2 Measure the angles of quadrilateral. EFGH. m∠E m∠F m∠G m∠H G E H Find the sum of the measures of the angles of quadrilateral EFGH. J 3 Measure the angles of quadrilateral. IJKL. m∠I m∠J m∠K m∠L K L I Find the sum of the measures of the angles of quadrilateral IJKL. Try This The measures of three angles of a quadrilateral are given. Predict the measure of the fourth angle. You may want to draw the quadrilateral to check your prediction. 1. 50°, 120°, 85° 2. 75°, 75°, 135° 3. 88°, 131°, 107° 9-4 Polygons 369 Explore It! Learn It! Summarize It! Practice It! Apply It! Draw Conclusions 4. What can you conclude about the sum of the measures of the angles of a quadrilateral?. Activity 2 You can use the fact that the sum of the measures of the angles of a triangle is 180° to find the sums of the measures of the angles of other polygons. Draw diagonals from any one vertex to the other vertices in each figure and fill in the table. Number of Number of 180° sum sides triangles per triangle 4 2 · 180° Sum of angle measures 2 · 180° = 360° · 180° 5 Find the missing angle from the polygon with the angles given. 5. 100°, 170°, 25°, 115°, x° 6. 145°, 55°, 117°, 159°, 90°, x° Draw Conclusions 7. Describe how you could find the sum of the measures of the angles of a heptagon. 370 9-4 Polygons Copyright © by Holt McDougal. All rights reserved. Try This Explore It! Learn It! Summarize It! Name Class Practice It! Apply It! 9-4 Date MA.8.G.2.3 Demonstrate that the sum of the angles in a triangle is 180 degrees...apply this fact to find unknown measure of angles, and the sum of angles in polygons Learn It! Polygons (Student Textbook pp. 413–417) Lesson Objective Classify and find angles in polygons Vocabulary polygon regular polygon trapezoid parallelogram rectangle rhombus Copyright © by Holt McDougal. All rights reserved. square Example 1 A. Find A Fi d the h sum of the angle measures in a hexagon. triangles. Divide the figure into · 180° = ° B. Find the sum of the angle measures in an octagon. triangles. Divide the figure into · 180° = ° Lesson Tutorial Videos @ thinkcentral.com 9-4 Polygons 371 Learn It! Explore It! Summarize It! Practice It! Apply It! Check It Out! 1. Find the sum of the angle measures in a heptagon. Example 2 Find the angle measures in each regular polygon. A. congruent angles 8x° = 180°(8 - 2) 8x° = 180°( 8x° = x° x° x° x° x° x° x° x° ) ° congruent angles y° y° y° y° 4y° = 180°(4 - 2) 4y° = 180°( ) ° 4y° = 4y° 360° _____ = _____ 8x° = _____ 1080° _____ ° ° y° = Check It Out! 2. Find the angle measures in the regular polygon. y° y° y° y° y° y° y° y° y° 372 9-4 Polygons Lesson Tutorial Videos @ thinkcentral.com Copyright © by Holt McDougal. All rights reserved. x° = B. Explore It! Learn It! Summarize It! Practice It! Apply It! Example 3 Give all Gi ll th the names that apply to each figure. A. -sided polygon 6 cm 6 cm 2 pairs of 6 cm 4 6 cm angles 4 sides 4 angles and sides 4 B. 2 ft J sided polygon K 2 ft 2 pairs of 2 ft M Check It Out! Copyright © by Holt McDougal. All rights reserved. 3a. 4 L 2 ft sides sides side Give all the names that apply to each figure. G B A D C AB || DC 3b. F 1 cm 1 cm G E 1 cm 1 cm H ___ ___ ___ ____ FG || EH, EF || HG Lesson Tutorial Videos @ thinkcentral.com 9-4 Polygons 373 Explore It! 9-4 Summarize It! Learn It! Name Practice It! Class Apply It! Date LA.8.2.2.3 The student will organize information to show understanding or relationships among facts…(e.g., representing key points…through… summarizing…) Summarize It! Polygons Think and Discuss 1. Choose which is larger, an angle in a regular heptagon or an angle in a regular octagon. Justify your answer. 2. Explain why all rectangles are parallelograms and why all squares are rectangles. 3. Explain why it is not possible to draw a diagonal of a triangle. Definition Examples 374 9-4 Polygons Property Polygon Nonexamples Copyright © by Holt McDougal. All rights reserved. 4. Get Organized Complete the graphic organizer. To fill in the sections, write the definition of a polygon and write a property of polygons. Then sketch examples and nonexamples of polygons. Explore It! Learn It! Summarize It! Name Practice It! Class Apply It! 9-4 Date MA.8.G.2.3 Demonstrate that the sum of the angles in a triangle is 180 degrees and apply this fact to find unknown measure of angles, and the sum of angles in polygons. Practice It! Polygons Find the sum of the angle measures in each polygon. 1. 2. 3. Find the angle measures in each regular polygon. 4. regular octagon 5. regular triangle 6. regular decagon Copyright © by Holt McDougal. All rights reserved. Honeybees store their honey in honeycombs. The honeycomb is made of many small wax compartments that are regular polygons as shown. 7. What type of polygon is each compartment? 8. What is the sum of the angle measures in each compartment? 9. What is the measure of each angle in any compartment? 10. A campground site is in the shape of a quadrilateral. Three sides of the campground form two right angles. The third interior angle measures 10° less than the fourth angle. Find the measure of each angle. 11. Three interior angles of a heptagon measure 125°, and two of the angles measure 143°. The heptagon has one right angle. What is the measure of the remaining angle? 12. Use the diagram to find the value of y. x° x° x° x° x° y° 9-4 Polygons 375 Explore It! 9-4 Learn It! Name Summarize It! Practice It! Class Apply It! Date MA.8.G.2.3 Demonstrate that the sum of the angles in a triangle is 180 degrees and apply this fact to find unknown measure of angles, and the sum of angles in polygons. Apply It! Polygons 1. A stop sign is a regular octagon. What is the sum of the angle measures in a stop sign? What is the measure of each of the sign’s angles? 2. One face of a gem is a quadrilateral with a 38° angle, a 142° angle, and a 110° angle. What is the measure of the remaining angle? 5. The Pentagon building in Arlington, Virginia, is a regular pentagon. Each side of the building is 921 feet long. About how long would it take to walk around the edge of the building at 350 ft/min? Round to the nearest tenth of a minute. 6. The figure shows the truss for a bridge. Suppose m∠BCD = 156° and ∠ABC ∠CDE. Find m∠ABC. C 3. In baseball, home plate is a pentagon with three right angles. The other two angles are congruent. What is the measure of each of the two congruent angles? 115° A R S 376 9-4 Polygons 115° E 7. To make string art, Eric places nails at each vertex of a regular hexagon. Then he strings a piece of colored wire along each diagonal of the hexagon. Along how many diagonals does he string the wire? Q P D 8. Gridded Response Jim is making a frame in the shape of a rhombus. One side is 18 cm long. When he buys the wood for the frame, he allows 10% extra for waste. How many centimeters of wood should he buy? Copyright © by Holt McDougal. All rights reserved. 4. Jasmine is making a kite as shown in the figure. She wants to make the kite so that m∠R is two times m∠P and m∠Q is three times m∠P. She also wants ∠S and ∠Q to be congruent. What measures must she use for each angle? B 9-1 Name Class Got It? THROUGH Date 9-4 Ready to Go On? Go to thinkcentral.com Quiz for Lessons 9-1 through 9-4 9-1 Angle Relationships (Student Textbook pp. 400–404) Use the diagram to name each figure. E 1. two pairs of complementary angles F D 2. three pairs of supplementary angles A 75° 55° 15° 35° B C 3. two right angles 9-2 Parallel and Perpendicular Lines (Student Textbook pp. 405–408) In the figure, line m || line n. Find the measure of each angle. Justify your answer. 125° 1 2 4. ∠1 3 4 m n 5. ∠3 9-3 Triangles (Student Textbook pp. 409–412) Copyright © by Holt McDougal. All rights reserved. Find x° in each triangle. 6. 60° 9-4 7. 83° 74° x° x° x° Polygons (Student Textbook pp. 413–417) 8. Give all of the names that apply to the figure. D A C AB || CD B Find the angle measures and the sum of the angle measures in each regular polygon. 9. a regular nonagon 10. a regular 11-gon Chapter 9 Geometry and Measurement 377 9-1 THROUGH Name 9-4 Class Date Connect It! MA.8.G.2.2; MA.8.G.2.3 Connect the concepts of Lessons 9-1 through 9-4 Angle Rummy Find a partner and follow these steps to play angle rummy. 1. Prepare a deck of 30 cards. Write the following angle measures on two index cards each: 10°, 20°, 30°, 40°, ... , 130°, 140°, 150° 2. The dealer shuffles the deck and deals 7 cards to each player. The remaining cards are placed face down in a stack. The player who is not the dealer goes first. 3. Play as follows: If you hold a set, place those cards face up on the table. If not, take a card from the top of the stack. If you can make a set after taking the card, place the set on the table. One set per turn! Sets 2 cards with complementary angles 2 cards with supplementary angles 3 cards with angles that form a triangle 4 cards with angles that form a quadrilateral 4. Alternate turns until a player holds no cards or when no cards remain in the stack. The player who has placed the most cards on the table is the winner. 5. Play the game a few times. Describe a helpful strategy. 1. Find the values of the variables d, g, s, and w. When a variable appears in more than one figure, it has the same value. 140 ° g° d° g° l || m 149 ° 36° l m w° s° 32° s° d = d° s = s° Think About The Puzzler 2. Which variable’s value did you find first? Why? 3. Can a quadrilateral have angle measures d°, g°, s°, and w°? Explain 378 Chapter 9 Geometry and Measurement g = w= Copyright © by Holt McDougal. All rights reserved. What’s Your Angle? Name Class 9-5 Date THROUGH 9-8 Remember It? Review skills and prepare for future lesson lessons. 9-5 Lesson Volume of Prisms and Cylinders (Student Textbook pp. 420–424) Rev. MA.7.G.2.1, Rev. MA.7.G.2.2 3 cm Find Fi d the h volume l to the nearest tenth. V = Bh = (πr2)h 4 cm = π(32)(4) = (9π)(4) = 36π cm3 ≈ 113.0 cm3 For 1–3, find the volume to the nearest tenth. Use 3.14 for π. 4.2 mm 13 cm 1. 2. 7 cm 7.5 mm 3. rectangular prism with length 8 ft, width 2 ft, and height 5 ft 8 cm Copyright © by Holt McDougal. All rights reserved. 4. A can has a diameter of 3 in. and a height of 5 in. Explain whether doubling the height would have the same effect on the volume as doubling the diameter. 9-6 Lesson Volume of Pyramids and Cones (Student Textbook pp. 425–429) Rev. MA.7.G.2.1 Find Fi d the h volume. l 7m V = __13Bh = __13(5)(6)(7) = 70 m3 6m 5m For 5–7, find the volume to the nearest tenth. Use 3.14 for π. 5. 6. 4.2 yd 7. 46 mm 8 in. 31 mm 31 mm 9 in. 5 in. Lesson Tutorial Videos @ thinkcentral.com 8 yd Chapter 9 Geometry and Measurement 379 9-7 Lesson Surface Area of Prisms and Cylinders (Student Textbook pp. 430–433) Rev. MA.7.G.2.1 Find Fi d the h surface f area of the rectangular prism. S = 2B + Ph 4 in. = 2(6) + (10)(4) = 52 in2 2 in. 3 in. For 8–11, find the surface area of each figure to the nearest tenth. Use 3.14 for π. 8. 9. 14.9 mm 21 ft 8 mm 20 mm 20 ft 10 mm 40 ft 12 mm 10. a rectangular prism with length 6 m, width 3 m, and height 3 m 11. a cylinder with radius 10 cm and height 5 cm 12. A tissue box has two square bases 11 cm on a side and a height of 14 cm. A pattern covers all surfaces. What is the area of the pattern? 9-8 Lesson Surface Area of Pyramids and Cones (Student Textbook pp. 434–437) Find Fi d the h surface f area of the pyramid. 4 in. 4 in. = 56 in2 For 13–16, find the surface area of each figure to the nearest tenth. Use 3.14 for π. 13. 14. 8 cm 6 cm 10 in. 15. 1 m 1m 6 cm 12 in. 16. a square pyramid with an 8 in. by 8 in. base and a height of 3 in. 17. Explain whether doubling the height and diameter of the cone in Exercise 15 would double the surface area. 380 Chapter 9 Geometry and Measurement Lesson Tutorial Videos @ thinkcentral.com Copyright © by Holt McDougal. All rights reserved. S = B + __12 Pl = 16 + __12(16)(5) Rev. MA.7.G.2.1 5 in. Explore It! Learn It! Summarize It! Name Practice It! Apply It! Class 9-9 Date MA.8.G.5.1 Compare, contrast, and convert units of measure between different measurement systems and dimensions including…area, volume, and derived units to solve problems. Explore It! Scaling Three-Dimensional Figures Scale Dimensions When you multiply each dimension of a two- or threedimensional figure by one number, you can predict how the area or volume will change. Activity 1 1 Complete the table. Find the area of each square. Square Side length (s) A B C D E F G 1 2 3 4 6 8 9 Area of a square with side length s in units2 s s 2 What patterns do you notice in the area of the squares? 3 The dimensions of square D are 2 times those of square B. How do the areas compare? Copyright © by Holt McDougal. All rights reserved. 4 The dimensions of square G are 3 times those of square C. How do the areas compare? Try This For 1–2, use the table in Activity 1. 1. Find another pair of squares where one’s dimensions are twice the other’s. How do these areas compare? 2. Find another pair of squares where one’s dimensions are 3 times the other’s. How do these areas compare? Draw Conclusions 3. When the dimensions of a square are multiplied by 2, the area is multiplied by . 4. When the dimensions are multiplied by 3, the area is multiplied by . 5. When the dimensions are multiplied by n, the area is multiplied by . 9-9 Scaling Three-Dimensional Figures 381 Explore It! Learn It! Summarize It! Practice It! Apply It! Activity 2 1 Complete the table. Find the volume of each cube. Cube Side length (s) A B C D E 1 2 3 4 6 F G Volume of a cube with side length s in units3 s s s 2 What patterns do you notice in the volumes of the cubes? 3 Notice that the dimensions of cube D are 2 times those of cube B. How do the volumes compare? 4 Notice that the dimensions of cube E are 3 times those of cube B. How do the volumes compare? Try This For 6–7, use the table in Activity 2. 6. Find another pair of cubes where one’s dimensions are 2 times the other’s. How do these volumes compare? Draw Conclusions 8. When the dimensions of a cube are multiplied by 2, the volume is multiplied by . 9. When the dimensions are multiplied by 3, the volume is multiplied by . 10. When the dimensions are multiplied by n, the volume is multiplied by . 382 9-9 Scaling Three-Dimensional Figures Copyright © by Holt McDougal. All rights reserved. 7. Find another pair of cubes where one’s dimensions are 3 times the other’s. How do these volumes compare? Learn It! Explore It! Summarize It! Name Practice It! Class Apply It! 9-9 Date MA.8.G.5.1 Compare, contrast, and convert units of measure between different measurement systems and dimensions including…area, volume, and derived units to solve problems. Learn It! Scaling Three-Dimensional Figures (Student Textbook pp. 438–441) Lesson Objective Make scale models of solid figures Vocabulary capacity Example 1 A 3 cm cube is built from small cubes, each 1 cm on an edge. Compare the following values. A. the edge lengths of the two cubes 3-cm cube ________ 1-cm cube cm ________ = Ratio of corresponding cm Copyright © by Holt McDougal. All rights reserved. The edges of the large cube are times as long as the edges of the small cube. B. the surface areas of the two cubes 3-cm cube ________ 1-cm cube ( 6( 2 ) cm ) cm 6 cm2 ___________= _________ = Ratio of corresponding 2 cm2 2 = The surface area of the large cube is times that of the small cube. C. the volumes of the two cubes 3-cm cube ________ 1-cm cube ( ( 3 ) cm ) cm cm3 __________ = _________ = 3 cm3 Ratio of corresponding 3 The volume of the large cube is = Lesson Tutorial Videos @ thinkcentral.com times that of the small cube. 9-9 Scaling Three-Dimensional Figures 383 Learn It! Explore It! Summarize It! Practice It! Apply It! Check It Out! An 8 cm cube is built from small cubes, each 2 cm on an edge. Compare the following values. 1a. the edge lengths of the two cubes 1b. the surface areas of the two cubes 1c. the volumes of the two cubes Example 2 A. What is the scale of the model? 6 in. ____ 4 ft 6 in. = _____ Convert and simplify. _______ in. The scale of the model is . B. What are the length and width of the model? length: __18 · 3 ft = __18 · in. = in. width: __18 · 2 ft = __18 · in. = in. 384 9-9 Scaling Three-Dimensional Figures Lesson Tutorial Videos @ thinkcentral.com Copyright © by Holt McDougal. All rights reserved. A box is in the shape of a rectangular prism. The box is 4 ft tall, and its base has a length of 3 ft and a width of 2 ft. For a 6 in. tall model of the box, find the following. Explore It! Learn It! Summarize It! Practice It! Apply It! Check It Out! A box is in the shape of a rectangular prism. The box is 5 ft tall, and its base has a length of 6 ft and a width of 4 ft. For a 6 in. tall model of the box, find the following. 2a. the scale of the model 2b. the length and width of the model Example 3 It takes 30 s for a pump to fill a cubic container whose edge measures 1 ft. How long does it take to fill a cubic container whose edge measures 2 ft? Vsmaller = 1 ft · 1 ft · 1 ft = 1 ft3 ft · Vlarger = Find the volume of each container. ft · ft = Copyright © by Holt McDougal. All rights reserved. xs 30 s = _______ ____ 1 ft3 ft3 Set up a proportion and solve. ft3 · =x Cross multiply. =x It takes s, or min, to fill the larger container. Check It Out! 3. It takes 8 s for a machine to fill a cubic box whose edge measures 4 cm. How long would it take to fill a cubic box whose edge measures 10 cm? Lesson Tutorial Videos @ thinkcentral.com 9-9 Scaling Three-Dimensional Figures 385 Explore It! 9-9 Learn It! Name Summarize It! Practice It! Class Summarize It! Apply It! Date LA.8.2.2.3 The student will organize information to show understanding or relationships among facts…(e.g., representing key points…through… summarizing…) Scaling Three-Dimensional Figures Think and Discuss 1. Describe how the volume of a model compares to the original object if the linear scale factor of the model is 1:2. 2. Explain one possible way to double the surface area of a rectangular prism. 3. Get Organized Complete the graphic organizer. To fill in the table, tell how the surface area and volume of a three-dimensional figure change when all of the figure’s dimensions are multiplied by each value. Scaling Three-Dimensional Figures The surface area is multiplied by... 2 3 4 5 1 2 1 3 2 5 n 386 9-9 Scaling Three-Dimensional Figures Then its volume is multiplied by... Copyright © by Holt McDougal. All rights reserved. All dimensions of a three-dimensional figure are multiplied by... Explore It! Learn It! Summarize It! Name Practice It! Practice It! Class Apply It! 9-9 Date MA.8.G.5.1 Compare, contrast, and convert units of measure between different measurement systems and dimensions including...area, volume, and derived units to solve problems. Scaling Three-Dimensional Figures A 16-cm. cube is built from small cubes, each 4 cm on a side. Find and compare the following values. 1. the edge lengths of the two cubes 2. the surface areas of the two cubes Copyright © by Holt McDougal. All rights reserved. 3. the volumes of the two cubes An oil drum is in the shape of a cylinder. The drum is 97 cm tall with a diameter of 58 cm. A model of the drum in a model railroad setup has a diameter of 9.0625 mm. Find the following to the nearest tenth. 4. the scale of the model 5. the height of the drum in the model 6. Find the volume of the drum in the model, to the nearest tenth. Use 3.14 for π. 7. A machine fills a 1 foot by 1 foot by 2 foot box with packing material in 12 seconds. How long would it take to fill a box with dimensions 3 times that of the smaller box? 8. A water pump can fill a cylindrical pool with a height of 4 feet and radius of 8 feet in 18 hours. How long would it take to fill a children’s pool with a height of 2 feet and a radius of 4 feet? 9-9 Scaling Three-Dimensional Figures 387 Explore It! 9-9 Learn It! Name Summarize It! Practice It! Class Apply It! Date MA.8.G.5.1 Compare, contrast, and convert units of measure between different measurement systems and dimensions including…area, volume, and derived units to solve problems. Apply It! Scaling Three-Dimensional Figures 1. John is designing a shipping container to hold boxes. The container he designs holds 24 boxes of the same size. Suppose he doubles the side lengths of the container. How many boxes will the larger container hold? 2. Maria uses two boxes of sugar cubes to create a solid building for a class project. She decides that the building is too small and that she will rebuild it so that each dimension is tripled. How many more boxes of sugar cubes will she need? 4. A building has a volume that is 125,000 times the volume of a model of the building. What is the ratio of the building’s dimensions to the model’s dimensions? 388 9-9 Scaling Three-Dimensional Figures Tank Height (ft) Radius (ft) A 12 6 B 10 12 5. It takes 6 cans of paint to paint the lateral surface of Tank A. Patrick is going to paint the lateral surface of Tank B. How many cans of paint should he buy? 6. Both water tanks are filled at the same rate. Tank A can be filled in 4 hours. How much longer does it take to fill Tank B? 7. Short Response A cereal box is 6 in. by 3 in. by 8 in. The price of the cereal is $1.85. The cereal is available in a larger box that is 9 in. by 4.5 in. by 12 in. What price would be proportional for the large box? Explain. Copyright © by Holt McDougal. All rights reserved. 3. It takes 8 minutes to drain Alicia’s aquarium. Carmen has an aquarium with each dimension 3 times that of Alicia’s aquarium. How long does it take to drain Carmen’s aquarium, assuming both aquariums drain at the same rate? Justify your answer. Express your answer in hours and minutes. The table shows the dimensions of two cylindrical water tanks. Use the table for 5−6. Explore It! Learn It! Summarize It! Name Practice It! Class Apply It! 9-10 Date MA.8.G.5.1 Compare, contrast, and convert units of measure between different measurement systems and dimensions including…area, volume, and derived units to solve problems. Explore It! Measurement in Three-Dimensional Figures Relate Measurement Stystems The basic units of length in the customary system are different from those in the metric system. You can find the relationship between them, however, and then extend the relationship to two and three dimensions. Activity 1 1 Use a metric ruler and an inch ruler. Measure the length and width of the rectangle in centimeters and in inches. Round to the nearest tenth. Length Width Customary Metric Copyright © by Holt McDougal. All rights reserved. 2 To find the number of centimeters in 1 inch, calculate the ratios metric length metric width _____________ and ____________ . Average the two results customary length customary width and round to hundredths. customary length customary width 3 Repeat Step 2 using the ratios _____________ and ____________. metric length metric width 1 in. ≈ cm 1 cm ≈ in. 4 Find the area of the rectangle in square inches and in square centimeters. customary area: in2 cm2 metric area: 5 Divide the metric area by the customary area to find the number of square centimeters in 1 square inch. Round to the nearest tenth. 1 in2 ≈ cm2 6 What relationship can you find between your results in Step 2 and Step 5? Try This Use the values you found above to convert between customary and metric units. Round to the nearest tenth. 1. 18 in. 2. 44 in. 3. 24 in2 4. 160 in2 5. 10 cm 6. 32.5 cm 7. 44 cm2 8. 115 cm2 9-10 Measurement in Three-Dimensional Figures 389 Explore It! Learn It! Summarize It! Practice It! Apply It! Activity 2 1 Look for a pattern in the conversions shown in the box at the right. Then use the pattern to make a conjecture about the relationship between the customary and metric systems with respect to volume. length: 1 in. = 2.54 cm area: 1 in2 = (2.54)2 cm2 = 6.45 cm2 volume: 1 in3 = ( )3 1 in3 = 2 To test your conjecture, find the volume of the cube in the customary and metric systems. 9 in. = customary volume: cm3 cm metric volume: 9 in. 9 in. metric volume 3 Use the volumes in Step 2 find the ratio _____________ . customary volume 9 in. 4 Compare your results in Step 3 with your conjecture in Step 1. What do you find? Try This 12 in. Find the area of the triangle in square centimeters to the nearest tenth two ways. 10. Find the area in square inches. 9. Convert each dimension to cm first. base = cm height = cm A = _12bh = in2 20 in. A = _12 bh = cm2 A= cm2 Find the volume of the prism in cubic inches to the nearest tenth two ways. 11. Convert each dimension to inches first. Then find volume. length ≈ V = lwh ≈ in. width ≈ in in. height ≈ 4 cm in. 3 12. Find the volume in cm3. Then convert to in3 using 1 in3 ≈ 16.39 cm3 V = lwh = cm3 V ≈ in3 Draw Conclusions 13. To convert surface area of a three dimensional figure between measurement systems, would you use the methods in Activity 1, Activity 2, or neither? Explain. 390 9-10 Measurement in Three-Dimensional Figures 3 cm 6 cm Copyright © by Holt McDougal. All rights reserved. Convert to cm2 using 1 in2 ≈ 6.45 cm2 Learn It! Explore It! Summarize It! Name Practice It! Apply It! Class 9-10 Date MA.8.G.5.1 Compare, contrast, and convert units of measure between different measurement systems and dimensions including...area, volume, and derived units to solve problems. Learn It! Measurement in Three-Dimensional Figures (Student Textbook pp. 442–445) Lesson Objective Convert units of measure in two and three dimensions within measurement systems and between different measurement systems. Example 1 A shoebox has a length of 13 inches, a width of 9 inches, and a height of 4__12 inches. Find the surface area of the box in square centimeters to the nearest tenth. Step 1: Find the surface area in square inches S=2· + lateral area S = 2 · (length)(width) + perimeter · ( + 9)(4__12 ) = 2 · (13)(9) + 2( ) Substitute = 234 + 198 = 432 in2 Copyright © by Holt McDougal. All rights reserved. Step 2: Find a conversion factor for inches to centimeters. 1 inch = cm, so the conversion factor is . Step 3: Convert the surface area. 54m 432 in ( ____ 1in. 2 2 ) = 2787.0912 cm2 The surface area of the shoebox is about . Check It Out! 1. A cone has h a radius d of 3 centimeters, a height of 4 centimeters, and a slant height of 5 centimeters. What is the surface area of the cone in square inches to the nearest tenth? Use 3.14 for π. Lesson Tutorial Videos 9-10 Measurement in Three-Dimensional Figures 391 Learn It! Explore It! Summarize It! Practice It! Apply It! Example 2 A standard beverage can is a cylinder with a radius of 3.25 cm and a height of 10.7 cm. What is the volume of the can in cubic inches to the nearest tenth? Step 1: Find the volume in centimeters. V = π r2h 2 ) · V = 3.14( Use 3.14 for π. V = 354.88 cm Step 2: Find a conversion factor for to 1 inch = 2.54 cm, so the conversion factor is Step 3: Convert the to . . . 1 in. 3= 21.6761063 354.88 cm3 ______ 2.54 cm ( ) The volume of the can is about . Check It Out! 2. Find 2 d the h approximate volume of the cone in cubic feet. 1m Copyright © by Holt McDougal. All rights reserved. 2m 392 9-10 Measurement in Three-Dimensional Figures Lesson Tutorial Videos Explore It! Learn It! Summarize It! Practice It! Apply It! Example 3 An archaeologist wants to apply a liquid solution to the lateral area of a square pyramid as a protectant. Each side of the square base measures 12 meters and the slant height is 10 meters. One gallon of solution covers 200 ft2. About how many gallons of solution does the architect need to cover the lateral area of the pyramid? Step 1: Find the lateral area in m2. Do not include the area of the base. L = __12 pℓ = __12 (4 · )(10) = 240 m2 Step 2: Find a conversion factor for meters to feet. m, so the conversion factor is 1 foot = . feet. Step 3: Convert to 240 m2 = 2583.3385 ft2, or about ft2 Step 4: Find the number of gallons needed. ft2 = 12.915. ft2 About gallons are needed. 3 ft Copyright © by Holt McDougal. All rights reserved. Check It Out! 3. The concrete tile shown is a hexagonal prism. A cubic yard of concrete 1 ft weighs about 3600 pounds. What is the weight of 40 tiles in tons. Lesson Tutorial Videos 2.6 ft 9-10 Measurement in Three-Dimensional Figures 393 Explore It! 9-10 Learn It! Summarize It! Name Practice It! Class Summarize It! Apply It! Date LA.8.2.2.3 The student will organize information to show understanding or relationships among facts… (e.g., representing key points…through… summarizing…) Measurement in Three-Dimensional Figures Think and Discuss 1. Explain how to convert an area in square feet to square centimeters. 2. Draw a diagram to show the number of cubic feet in a cubic yard. 3. Get Organized Complete the graphic organizer. To fill in the boxes, describe the steps for finding the volume of a prism in cubic meters when the dimensions of the prism are given in yards. You may use any number of steps. 1. 2. 3. Volume of the prism given in cubic meters 394 9-10 Measurement in Three-Dimensional Figures Copyright © by Holt McDougal. All rights reserved. Dimensions of a prism given in cubic yards Explore It! Learn It! Practice It! Summarize It! Name Apply It! Class 9-10 Date MA.8.G.5.1 Compare, contrast, and convert units of measure between different measurement systems (US customary or metric (SI)) and dimensions including... area, volume, and derived units to solve problems. Practice It! Measurement in Three-Dimensional Figures 1. Find the approximate area in in2. 2. Find the approximate area in cm2. 3. Find the approximate area in m2. 6 in. 160 ft 80 ft 20 cm 4 in. 20 cm 86 ft Use the cylinder at the right for 4–5. 15 mm 4. Find the approximate surface area in in2. 10 mm Copyright © by Holt McDougal. All rights reserved. 5. Find the approximate volume in in3. 6. Find the approximate volume of the triangular prism in cubic centimeters. 2 in. 1 in. 6 in. 7. A store sells tea leaves in cylindrical containers. A certain tea costs $0.25 per teaspoon. What is the cost of filling the container with tea? 1 in3 ≈ 3.3 tsp. 3.1 cm 12 cm 8. A cereal box is a rectangular prism that is 40 cm tall, 30 cm wide, and 5 cm thick. A new box will have the same volume, a width of 12 inches and a thickness of 2 inches. a. How tall is the new box in inches? b. Is this taller or shorter than the original box? Justify your answer. 9-10 Measurement in Three-Dimensional Figures 395 Explore It! 9-10 Learn It! Summarize It! Name Apply It! Practice It! Class Date MA.8.G.5.1 Compare, contrast, and convert units of measure between different measurement systems (US customary or metric (SI)) and dimensions including…area, volume, and derived units to solve problems. Apply It! Measurement in Three-Dimensional Figures 1. A crate in the shape of a rectangular prism with linear dimensions 1 yd by _12 yd by _3 yd is being filled with topsoil. How many 4 gallons of topsoil will fit? 1 in3 ≈ 0.0043 gal 5. Construction paper was used to make a model of a pyramid for a class project. The dimensions are shown. 4m 2. The diagram shows a stage floor. 2m 3m 28 ft Find the volume in cubic feet. 26 ft 40 ft Estimate the cost to varnish the stage floor if the cost is $1.99 per square meter. 4. The glass cone below is to be filled with colored sand at $0.02 per tablespoon. If 1 cm3 is about 0.0676 tbsp, how much will the sand cost? (Hint: Convert cubic inches to cubic centimeters to tablespoons.) 3 in. 7. Gridded Response Nora is measuring fabric in cubic yards. The price is given per cubic meter. She knows that 1 yard is 0.9144 meter. What number completes the conversion factor she will use? Round to three decimal places. m3 1 yd3 = _________ 3 1 yd 6 in. 396 9-10 Measurement in Three-Dimensional Figures Copyright © by Holt McDougal. All rights reserved. 3. Water covers 5983 square miles of Florida and land covers 53,927 square miles. Find the difference between the land and water areas, in square kilometers. 6. Air passes through an air purifier at a rate of 30 m3/h. How long would it take for the purifier to process the air in a room with dimensions 12 ft by 20 ft by 8 ft? 9-9 Name Class Got It? THROUGH Date 9-10 Ready to Go On? Go to thinkcentral.com Quiz for Lessons 9-9 through 9-10 9-9 Scaling Three-Dimensional Figures (Student Textbook pp. 438–441) A cube 12 inches on a side is made up of small cubes 4 inches on a side. Find and compare the following values. 1. the side lengths of the two cubes 2. the surface areas of the two cubes 4. A skating arena is 400 ft long, 280 feet wide, and 100 feet tall. The scale model used to build the arena is 20 inches long. Find the width and height of the model. 5. A cube 6.5 inches on a side has a cube 5 inches on a side inside it. Which volume is greater, the remaining space inside the cube or the volume of the small cube? By what percent is the volume greater? 9-10 Measurement in Three Dimensional Figures (Student Textbook pp. 442–445) Find the area of a square with side length 130 centimeters to the nearest tenth. 6. in square meters 7. in square feet Find the volume of the cone to the nearest tenth. Use 3.14 for π. 5 ft Copyright © by Holt McDougal. All rights reserved. 3. the volumes of the two cubes 8. in cubic inches 9 ft 9. in cubic meters 10. An industrial popcorn making machine makes 175 1-ounce servings per hour. One ounce is about 2000 cm3. How long would it take to fill a 24 inch by 20 inch by 36 inch popcorn cart? Chapter 9 Geomentry and Measurement 397 9-9 Name THROUGH 9-10 Class Date Connect It! MA.8.G.2.4; MA.8.G.5.1 Connect the concepts of Lessons 9-9 through 9-10 A Fishy Problem Javier is looking for an aquarium that fits in the corner of a room. He wants an aquarium that holds at least 20 gallons of water. 1. Javier sees the aquarium at right in a store. The base of the aquarium is an isosceles right triangle. Write and solve an equation to find the length x to the nearest centimeter. 180 cm 2. Find the volume of the aquarium. x 3. Show how to convert the volume to cubic feet. (Hint: 1 ft = 30.48 cm) x 42 cm 4. One cubic foot is approximately 7.48 gallons. Should Javier buy the aquarium? Why or why not? In the Thick of It 1. In each box, circle the letter of the measurement that is closer to the measurement shown in blue. 600 in2 2 120 m2 2 2 8 in3 2 50 ft 4 ft 1292 ft 394 ft 131 cm A R T N O 3 1600 ft3 3 52 cm 149 m I S 3 2. Arrange the circled letters to find the animal that has up to a million hairs per square inch. Think About The Puzzler 3. Explain how you can use estimation to help solve the puzzle. 398 Chapter 9 Geometry and Measurement 45 m T 390 cm3 3 24 in E 3 60 in H 3 Copyright © by Holt McDougal. All rights reserved. Which animal has the thickest fur? Solve the puzzle to find out! FLORIDA Name Class Study It! Vocabulary CHAPTER Date 9 Multi-Language Glossary Go to thinkcentral.com (Student Textbook page references) acute angle . . . . . . . . . . . . . (400) obtuse angle . . . . . . . . . . . . (400) scalene triangle . . . . . . . . . (410) acute triangle . . . . . . . . . . . (409) parallel lines. . . . . . . . . . . . (406) slant height. . . . . . . . . . . . . (434) adjacent angles . . . . . . . . . (401) parallelogram . . . . . . . . . . (415) square . . . . . . . . . . . . . . . . . (415) angle. . . . . . . . . . . . . . . . . . . (400) perpendicular lines . . . . . (406) straight angle . . . . . . . . . . . (400) capacity . . . . . . . . . . . . . . . . (438) polygon . . . . . . . . . . . . . . . . (414) surface area . . . . . . . . . . . . (430) complementary angles . . (400) rectangle . . . . . . . . . . . . . . . (415) supplementary angles . . . (400) congruent angles . . . . . . . (401) regular polygon . . . . . . . . . (415) transversal . . . . . . . . . . . . . (406) equilateral triangle. . . . . . (410) regular pyramid . . . . . . . . (434) trapezoid . . . . . . . . . . . . . . . (415) isosceles triangle. . . . . . . . (410) rhombus . . . . . . . . . . . . . . . (415) Triangle Sum Theorem . . (402) lateral face . . . . . . . . . . . . . (430) right angle. . . . . . . . . . . . . . (400) vertical angles . . . . . . . . . . (401) lateral surface . . . . . . . . . . (430) right cone . . . . . . . . . . . . . . (434) volume . . . . . . . . . . . . . . . . . (420) obtuse triangle. . . . . . . . . . (409) right triangle . . . . . . . . . . . (409) Complete the sentences below with vocabulary words from the list above. 1. Lines in the same plane that never meet are called . Copyright © by Holt McDougal. All rights reserved. Lines that intersect at 90° angles are called . 2. A quadrilateral with 4 congruent angles is called a . A quadrilateral with 4 congruent sides is called a . 3. The nonadjacent angles formed by two intersecting lines are called Lesson 9-1 . Angle Relationships (Student Textbook pp. 400–404) A Find the angle measure. m∠1 m∠1 + 122° = 180° 122° -122° −−−−−−−− −−−− m∠1 = 58° 1 2 MA.8.G.2.2 122° 3 Find each angle measure. 4. m∠1 5. m∠2 3 2 68° 1 6. m∠3 Lesson Tutorial Videos @ thinkcentral.com Chapter 9 Geometry and Measurement 399 Lesson 9-2 Parallel and Perpendicular Lines (Student Textbook pp. 405–408) P Line j line k. Find each angle measure. m∠1 m∠1 = 143° MA.8.G.2.2 Alternate interior angles are . 143° 2 m∠2 m∠2 + 143° = 180° - 143° - 143° −−−−−−−− −−−−− m∠2 = 37° 1 k j Line p line q. Find each angle measure. q 7. m∠1 8. m∠2 9. m∠3 10. m∠4 p 1 114° 3 2 4 5 11. m∠5 Lesson 9-3 Triangles (Student Textbook pp. 409–412) T MA.8.G.2.3 Find n°. n° + 50° + 90° n° + 140° - 140° −−−−−−−−−− n° = = 180° 180° - 140° −−−−− = 40° n° 50° 13. Find p°. m° p° 128° Lesson 9-4 m° 47° Polygons (Student Textbook pp. 413–417) P 78° MA.8.G.2.3 Find the angle measures in a regular 12–gon. 12x° = 180°(12 - 2) 12x° = 180°(10) 12x° = 1800° x° = 150° Find the angle measures in each regular polygon. 14. a regular hexagon 400 Chapter 9 Geometry and Measurement 15. a regular 11–gon Lesson Tutorial Videos @ thinkcentral.com Copyright © by Holt McDougal. All rights reserved. 12. Find m°. Lesson 9-9 Scaling Three-Dimensional Figures (Student Textbook pp. 438–441) S A 4-in. cube is built from small cubes, each 2 in. on a side. Compare the volumes of the large cube and the small cube. vol. of large cube ____________ = vol. of small cube MA.8.G.5.1 3 64 in 4 in 3 = _____ =8 ( ___ 2 in ) 8 in 3 The volume of the large cube is 8 times that of the small cube. A 9-ft cube is built from small cubes, each 3 ft on a side. Compare the indicated measures of the large cube and the small cube. 16. side lengths 17. surface areas 18. volumes Copyright © by Holt McDougal. All rights reserved. 19. A rectangular prism that is 4 in. deep, 12 in. wide, and 16 in. long is scaled down so that the new prism is 3 in. wide. How do the surface area and volume compare to the original surface area and volume? Lesson 9-10 Measurement in Three-Dimensional Figures M MA.8.G.5.1 ((Student Textbook pp. 442–445) How many square meters of sheet metal was used to construct the water tank to the nearest tenth?Use 3.14 for π. S = 2π(4)2 + 2π(4)(30) = 32π + 240 π = 272π = 854.08 ft2 1m 2 ≈ 79.4 m2 = 854.08 ft2 _____ 3.28 ft ( 4 ft ) 2 About 79.4 m of sheet metal was used to construct the water tank. 30 ft 20. What is the volume of the water tank above in cubic feet? in cubic meters? 21. If the tank fills at a rate of 3 gallons per second and 1 cubic meter is about 264 gallons, how long does the tank take to fill in minutes? Lesson Tutorial Videos @ thinkcentral.com Chapter 9 Geometry and Measurement 401 Name Class Write About It! Date LA.8.3.1.2 The student will prewrite by making a plan for writing that addresses purpose, audience, main idea, logical sequence, and time frame for completion. Think and Discuss Answer these questions to summarize the important concepts from Chapter 9 in your own words. 1. If m∠1 is 50°, explain how to find m∠5. 2 3 7 1 4 5 6 8 2. If the measures of two angles in a triangle are 43° and 74°, explain how to find the measure of the third angle. 4. If the side lengths of a rectangular prism are measured in centimeters, its surface area is A cm2 and its volume is V cm3, what is its surface area in in2 and volume in in3? Before The Test I need answers to these questions: 402 Chapter 9 Geometry and Measurement Copyright © by Holt McDougal. All rights reserved. 3. Explain how triangles help you find the angle measures in a regular polygon with more than 3 sides.