Chapter Quadrilaterals

Transcription

1 6 hapter Quadrilaterals How do photographers reach high places? Scissors lifts can lift photographers, movie crews, and other people who need to reach high places. The design of the lift ensures that as the platform is raised or lowered it is always parallel to the ground. Learn More bout It You will learn more about scissors lifts in Exercises on p

2 Who uses Quadrilaterals? FURNITURE ESIGNER Furniture designers use geometry and trigonometry to create designs for furniture that is structurally sound and visually interesting. (p. 329) GEMOLOGIST Gemologists consider the color and clarity of a gem, as well as the cut, when evaluating its value. fter cutting, the facets of some gems are rectangles and trapezoids. (p. 340) How will you use these ideas? iscover plant roots and stems that resemble polygons. (p. 307) See how photographers take overhead photos. (p. 314) Find out how the derailleur of a bicycle uses quadrilaterals. (p. 321) redict the path of a ball on a pool table. (p. 322) Learn how gems are cut to enhance their sparkle. (p. 340) 301

3 hapter 6 Study Guide REVIEW What s the chapter about? lassifying polygons and finding angle measures of quadrilaterals Using properties of parallelograms, rhombuses, rectangles, squares, and trapezoids Identifying special quadrilaterals Key Words polygon, p. 303 diagonal of a polygon, p. 303 parallelogram, p. 310 rhombus, p. 325 rectangle, p. 325 square, p. 325 trapezoid, p. 332 isosceles trapezoid, p. 332 midsegment of a trapezoid, p. 333 RERE hapter Readiness Quiz Take this quick quiz. If you are unsure of an answer, look at the reference pages for help. Vocabulary heck (refer to p. 173) 1. What type of triangle is TFGH? scalene equilateral isosceles None of these F G H Skill heck (refer to pp. 131, 180) 2. What is the measure of a4? F 45 G H 65 J In TJKL, maj 55 and mak 90. What is mal? VISUL STRTEGY Visualize It! rawing Quadrilaterals You can used lined paper to help you draw quadrilaterals that have at least one pair of opposite sides parallel. trapezoid parallelogram 302 hapter 6 Quadrilaterals

4 6.1 olygons Goal Identify and classify polygons. Find angle measures of quadrilaterals. Each traffic sign below is an example of a polygon. Notice that each sign is formed with straight lines. Key Words polygon side of a polygon vertex of a polygon diagonal of a polygon Stud t H lp VOULRY TI side connects consecutive vertices. diagonal connects nonconsecutive vertices. polygon is a plane figure that is formed by three or more segments called sides. Each side intersects exactly two other sides at each of its endpoints. Each endpoint is a vertex of the polygon. Two vertices that are the endpoints of the same side are called consecutive vertices. For example, in polygon QRST, R and S are consecutive vertices. segment that joins two nonconsecutive vertices of a polygon is called a diagonal. olygon QRST has two diagonals from vertex R, R &* and RT&*. diagonals T R vertex side vertex S R T S EXMLE 1 Identify olygons Is the figure a polygon? Explain your reasoning. a. b. c. d. Solution a. Yes. The figure is a polygon formed by four straight sides. b. No. The figure is not a polygon because it has a side that is not a segment. c. No. The figure is not a polygon because two of the sides intersect only one other side. d. Yes. The figure is a polygon formed by six straight sides. 6.1 olygons 303

5 lassifying olygons You can classify polygons by the number of sides they have. Some special types of polygons are listed below. Student Help STUY TI To classify a polygon not listed in the table, use the number of sides. For example, a polygon with 14 sides is a 14-gon. polygon with n sides is an n-gon. TYES OF OLYGONS Triangle Quadrilateral entagon 3 sides 4 sides 5 sides Hexagon Heptagon Octagon 6 sides 7 sides 8 sides EXMLE 2 lassify olygons ecide whether the figure is a polygon. If so, tell what type. If not, explain why. a. b. c. d. Solution a. The figure is a polygon with four sides, so it is a quadrilateral. b. The figure is not a polygon because it has some sides that are not segments. c. The figure is a polygon with five sides, so it is a pentagon. d. The figure is not a polygon because some of the sides intersect more than two other sides. Identify and lassify olygons ecide whether the figure is a polygon. If so, tell what type. If not, explain why hapter 6 Quadrilaterals

6 Quadrilaterals diagonal of a quadrilateral divides it into two triangles, each with angle measures that add up to 180. So, the sum of the measures of the interior angles of a quadrilateral is 2 180, or ma1 ma2 ma ma4 ma5 ma6 180 THEOREM 6.1 Quadrilateral Interior ngles Theorem Words The sum of the measures of the interior angles of a quadrilateral is Symbols ma1 ma2 ma3 ma Student Help STUY TI Name a polygon by listing its vertices consecutively in either direction. Two names for the quadrilateral in Example 3 are QRS and RQS. EXMLE 3 Find ngle Measures of Quadrilaterals Find the measure of as. Solution 80 Use the fact that the sum of the measures of the interior angles of a quadrilateral is 360. Quadrilateral Interior ngles ma maq mar mas 360 Theorem mas 360 Substitute angle measures R S 220 mas 360 mas 140 Simplify. Subtract 220 from each side. NSWER The measure of as is 140. Find ngle Measures of Quadrilaterals Find the measure of a olygons 305

7 6.1 Exercises Guided ractice Vocabulary heck 1. What type of polygon has 8 sides? 15 sides? 2. Use the diagram of the pentagon shown at the right. Name all of the diagonals from vertex. E Skill heck Is the figure a polygon? Explain your reasoning Find the measure of a ractice and pplications Extra ractice See p lassifying olygons ecide whether the figure is a polygon. If so, tell what type. If not, explain why Homework H lp Example 1: Exs Example 2: Exs. 8 10, 21, Example 3: Exs , Logical Reasoning What is the fewest number of sides a polygon can have? Explain your answer, then name the polygon. Visualize It! 12. Two different pentagons Sketch the figure(s) described. 13. hexagon with three diagonals drawn from a single vertex 14. quadrilateral with two obtuse angles 306 hapter 6 Quadrilaterals

8 Finding ngle Measures Find the measure of a IStudent Help I LSSZONE.OM HOMEWORK HEL Extra help with problem solving in Exs is at classzone.com Using lgebra Find the value of x (x 30) 3x 100 2x 84 2x arachutes Some gym classes play games using parachutes that look like the polygon below. 21. Tell how many sides the polygon has and what type of polygon it is. 22. olygon LMNQRST is one name for the polygon. State two other names using the vertices. T S L M N lants 23. Name all of the diagonals that have vertex M as an endpoint. Not all of the diagonals are shown. lants Use the following information. ross sections of roots and stems often resemble polygons. Next to each cross section is the polygon it resembles. Tell how many sides each polygon has and tell what type of polygon it is. Source: The History and Folklore of North merican Wildflowers R 24. Virginia Snakeroot 25. araway RMOL, or star fruit, has a cross section shaped like a five-pointed star. 26. Fennel 27. oison Hemlock 6.1 olygons 307

9 28. Technology Use geometry software to draw a quadrilateral. Measure each interior angle and calculate the sum. What happens to the sum as you drag the vertices of the quadrilateral? Standardized Test ractice 29. Multi-Step roblem Envelope manufacturers fold a speciallyshaped piece of paper to make an envelope, as shown below a. How many sides are formed by the outer edges of the paper before it is folded? Name the type of polygon. b. Tell how many sides are formed by the outer edges of the paper in Steps 2 4. Name the type of polygon formed after each step. c. If the four angles of the red quadrilateral in Step 4 are congruent, then what is the measure of each angle? Mixed Review Line Relationships etermine whether the lines are parallel, perpendicular, or neither. (Lesson 3.1) 30. ^&*( and E ^&*( 32. ^&*( and ^&*( 31. ^&*( and E ^&*( 33. ^&*( and E ^&*( E Finding ngle Measures Find the measure of the numbered angle. (Lesson 3.4) lgebra Skills istributive roperty Use the distributive property to rewrite the expression without parentheses. (Skills Review, p. 671) 37. 4(x 3) 38. (x 1) (x 7) 308 hapter 6 Quadrilaterals 40. 5(2x 3) 41. 3(5x 2) 42. (4x 4)x