Angle Measures and Angle Bisectors

Transcription

1 OON OE U T Z Y U T Locker LEON ommon ore ath tandards The student is expected to: OON OE G-O..1 Know precise definitions of angle... based on the undefined notions of distance around a circular arc. lso G-O..12 athematical ractices OON OE 16.2 ngle easures and ngle isectors.5 Using Tools Language Objective Work with a partner to play angle charades. ENGGE Essential uestion: How is measuring an angle similar to and different from measuring a line segment? ossible answer: In both cases, the measure is undefined until a unit is chosen. ngles may be measured in degrees; there are 360 in a circle. The tool for measuring an angle in degrees is a protractor. Line segments are measured using linear units, such as centimeters or inches. The tool for measuring a line segment is a ruler. EVIEW: LEON EFONE TK View the online Engage. iscuss the photo and the fact that 60 and 40 stands are available, but that a customer wants a 50 stand. Then preview the Lesson erformance Task. Name lass ate 16.2 ngle easures and ngle isectors Essential uestion: How is measuring an angle similar to and different from measuring a line segment? Explore onstructing a opy of an ngle tart with a point and use a compass and straightedge to construct a copy of. Use a straightedge to draw a ray with endpoint. lace the point of your compass on and draw an arc that intersects both sides of the angle. Label the points of intersection T and U. Without adjusting the compass, place the point of the compass on and draw an arc that intersects the ray. Label the intersection Y. U T E F lace the point of the compass on T and open it to the distance TU. Without adjusting the compass, place the point of the compass on Y and draw an arc. Label the intersection with the first arc Z. Use a straightedge to draw Z. is a copy of. eflect 1. If you could place the angle you drew on top of so that Y coincides with T, what would be true about Z? Explain. Z would coincide with U. ince the angles are copies of each other, the rays in each angle form the same opening. esource Locker 2. iscussion Is it possible to do the construction with a compass that is stuck open to a fixed distance? Why or why not? No; you could use the compass to make the required arcs in teps and, but you would not be able to adjust the opening of the compass as required in tep. odule Lesson 2 Name lass ate 16.2 ngle easures and ngle isectors Essential uestion: How is measuring an angle similar to and different from measuring a line segment? G-O..1 Know precise definitions of angle based on the undefined notions of distance around a circular arc. lso G-O..12 Explore onstructing a opy of an ngle tart with a point and use a compass and straightedge to construct a copy of. Use a straightedge to draw a ray with endpoint. lace the point of your compass on and draw an arc that intersects both sides of the angle. Label the points of intersection T and U. Without adjusting the compass, place the point of the compass on and draw an arc that intersects the ray. Label the intersection Y. lace the point of the compass on T and open it to the distance TU. esource Without adjusting the compass, place the point of the compass on Y and draw an arc. Label the intersection with the first arc Z. Use a straightedge to draw Z. is a copy of. Z U Y T HOVE GE Turn to these pages to find this lesson in the hardcover student edition. eflect 1. If you could place the angle you drew on top of so that Y coincides with T, what would be true about Z? Explain. Z would coincide with U. ince the angles are copies of each other, the rays in each angle form the same opening. No; you could use the compass to make the required arcs in teps and, but you would not be able to adjust the opening of the compass as required in tep. 2. iscussion Is it possible to do the construction with a compass that is stuck open to a fixed distance? Why or why not? odule Lesson Lesson 16.2

2 Explain 1 Naming ngles and arts of an ngle n angle is a figure formed by two rays with the same endpoint. The common endpoint is the vertex of the angle. The rays are the sides of the angle. Example 1 raw or name the given angle. ELOE onstructing a opy of an ngle When an angle is named with three letters, the middle letter is the vertex. o, the vertex of angle is point. The sides of the angle are two rays with common endpoint. o, the sides of the angle are and. raw and label the angle as shown. The vertex of the angle shown is point K. name for the angle is K. The vertex must be in the middle, so two more names for the angle are and L K J. The angle is numbered, so another name is 1. eflect L 3. Without seeing a figure, is it possible to give another name for KG? If so, what is it? If not, why not? Yes; GK Your Turn Use the figure for Name 2 in as many different ways as possible. E, E 1 K 5. Use a compass and straightedge to copy E. J 2 3 E 4 J K L odule Lesson 2 OFEIONL EVELOENT ath ackground ompass and straightedge constructions date to ancient Greece. In fact, one of the classic problems of ancient Greek mathematics was the trisection of an angle. That is, using a compass and straightedge, is it possible to construct an angle whose measure is one-third that of an arbitrary given angle? It was not until 1837 that this construction was proven to be impossible. On the other hand, it is a straightforward task to bisect any angle, and students learn this fundamental construction in this lesson. UETIONING TTEGIE o the rays of the angle you construct need to be the same length as the rays of the given angle? Why or why not? No; the measure of the angle is determined only by the size of the opening between the rays, not by the lengths of the rays. When you draw the initial arc that intersects the side of the angle to be copied, does it matter how wide you open the compass? Explain. No, as long as the arc intersects both sides of the angle, it doesn t matter. INTEGTE THETIL TIE Focus on odeling.4 Have students practice constructing both acute and obtuse angles. ELIN 1 Naming ngles and arts of an ngle ONNET VOULY onnect the word degree to the idea of measurement. degree in science may be a measure of temperature in units known as Fahrenheit or elsius. egree in this context is the measure of an angle. sk how many degrees are in a straight angle, a right angle, and so on. ngle easures and ngle isectors 790

3 UETIONING TTEGIE When an angle is named using three letters, how can you identify the vertex of the angle? The vertex is the center letter of the angle name. n angle diagram may use letters or numbers to identify the angle. How are the diagrams different? Letters label individual points on the angle, while a number is inside the angle and names the entire angle. Explain 2 easuring ngles The distance around a circular arc is undefined until a measurement unit is chosen. egrees ( ) are a common measurement unit for circular arcs. There are 360 in a circle, so an angle that measures 1 is 1 of a circle. 360 The measure of an angle is written m or m. You can classify angles by their measures. lassifying ngles cute ngle ight ngle Obtuse ngle traight ngle 0 < m < 90 m = < m < 180 m = 180 ELIN 2 easuring ngles Example 2 53 Use a protractor to draw an angle with the given measure. tep 1 Use a straightedge to draw a ray, Y. Y VOI OON EO emind students to place the center mark of the protractor on the vertex and to align one side of the angle with the 0 mark. They may have to rotate the angle or the protractor for ease of alignment. On some protractors, the zero line is on the bottom edge, while on others, it is placed higher. INTEGTE THETIL TIE Focus on odeling.4 uggest that students use a straightedge, such as an index card, to extend the rays of an angle before they use a protractor to measure the angle. If the angle is smaller than the distance from the center mark to the edge of the protractor, this will make it easier to accurately measure the angle. Encourage students to estimate an angle measure before measuring to make sure the measurement is reasonable. tep 2 lace your protractor on point as shown. Locate the point along the edge of the protractor that corresponds to 53. ake a mark at this location and label it point Z. tep 3 raw Z. m ZY = 53. Z odule Lesson 2 OLLOTIVE LENING mall Group ctivity Use pictures from magazines to find angles of different sizes. sk students to identify the type of angle and estimate the measure. Then have students measure the angles with a protractor. If protractors are not available, they can use index cards or origami paper. The edges are already at a 90 angle, and anything greater would be an obtuse angle. half-fold forms a 45 angle, a tri-fold approximately 30, and so on. The pictures can be posted by classification and used for reference. Z Y Y 791 Lesson 16.2

4 138 tep 1 Use a straightedge to draw a ray,. tep 2 lace your protractor on point so that is at zero. tep 3 Locate the point along the edge of the protractor that corresponds to 138. ake a mark at this location and label it point. tep 4 raw. m = 138. eflect 6. Explain how you can use a protractor to check that the angle you constructed in the Explore is a copy of the given angle. easure the given angle and the constructed angle. They should have the same measure. Your Turn Each angle can be found in the rigid frame of the bicycle. Use a protractor to find each measure N Explain 3 onstructing an ngle isector n angle bisector is a ray that divides an angle into two angles that both have the same measure. In the figure, bisects, so m = m. The arcs in the figure show equal angle measures. ostulate 2: ngle ddition ostulate If is in the interior of, then m = m + m K L J Image redits: Gena73/ hutterstock UETIONING TTEGIE If the vertex of an angle is placed on the center point of a protractor and both rays of the angle lie within the measures of the protractor, does one of the rays have to align with the 0 mark to find the measure of the angle? Explain. No, you can find the absolute value of the difference of the measures each ray intersects to find the measure of the angle. For example, if one ray aligns with 25 and the other with 67, the angle measures 42. ELIN 3 onstructing an ngle isector ONNET VOULY The postulates for angles are similar to the postulates for segments. The rotractor ostulate is similar to the uler ostulate. It says that the measure of an angle is the absolute value of the difference between the numbers matched on a protractor with the rays that form the sides of the angle. odule Lesson 2 IFFEENTITE INTUTION anipulatives Have students investigate how to find the bisector of an angle using a geometric reflecting tool. Have students draw an angle on a piece of paper. To use the tool, place it on the vertex of the angle so that one side is reflected onto the other side. Then draw the tool s line. iscuss how using the reflective device is similar to using paper folding to find the angle bisector. ngle easures and ngle isectors 792

5 VOI OON EO emind students not to change the compass setting when they draw the intersecting arcs from each side ray of an angle to create the angle bisector. In order to help students see why this is important, you many want to have them do a construction in which they change the compass setting between arcs. tudents will see that the resulting ray does not bisect the angle. Example 3 Use a compass and straightedge to construct the bisector of the given angle. heck that the measure of each of the new angles is one-half the measure of the given angle. tep 1 lace the point of your compass on point. raw an arc that intersects both sides of the angle. Label the points of intersection and. tep 2 lace the point of the compass on and draw an arc in the interior of the angle. UETIONING TTEGIE If a ray divides an angle into two angles with equal measures, what must be true about the ray? Explain. The ray is the angle bisector of the angle by the definition of an angle bisector. tep 3 Without adjusting the compass, place the point of the compass on and draw an arc that intersects the last arc you drew. Label the intersection of the arcs. tep 4 Use a straightedge to draw. VIUL UE ome students may have difficulty visualizing two angles that have the same measure, especially if the sides of the angles are shown with rays of different lengths. You may want to have students construct angle copies on tracing paper. Then they can place the copy on top of the original angle to check that the measures are the same. tep 5 easure with a protractor to confirm that m = m = 1_ 2 m. 27 = 27 = 1_ 2 (54 ) tep 1 raw an arc centered at that intersects both sides of the angle. Label the points of intersection and. tep 2 raw an arc centered at in the interior of the angle. tep 3 Without adjusting the compass, draw an arc centered at that intersects the last arc you drew. Label the intersection of the arcs. tep 4 raw. tep 5 heck that m = m = 1_ 2 m. Yes; 45 = 45 = 1_ 2 (90 ) odule Lesson 2 LNGUGE UOT onnect Vocabulary emind students that the prefix bi- means two and that the root sect means to cut. They can use these cues to help them remember that an angle bisector divides the angle into two equal parts. 793 Lesson 16.2

6 eflect 9. iscussion Explain how you could use paper folding to construct the bisector of an angle. Fold the paper so that one side of the angle lies on top of the other. Unfold the paper. The crease is the angle bisector. Your Turn Use a compass and straightedge to construct the bisector of the given angle. heck that the measure of each of the new angles is one-half the measure of the given angle ELOTE INTEGTE THETIL TIE Focus on ath onnections.1 emind students to record angle measures using a protractor in degrees by using the degree symbol. oint out that not all angle measures are recorded in degrees. adians are real number units of angle rotation. For example, π radians = 180. Elaborate 12. What is the relationship between a segment bisector and an angle bisector? segment bisector divides a line segment into two segments that have the same length; an angle bisector divides an angle into two angles that have the same measure. 13. When you copy an angle, do the lengths of the segments you draw to represent the two rays affect whether the angles have the same measure? Explain. No; the measure of an angle depends only on the portion of a circle that the angle encompasses, not upon the apparent length of its sides. 14. Essential uestion heck-in any protractors have two sets of degree measures around the edge. When you measure an angle, how do you know which of the two measures to use? nswers may vary. ample: First determine if the angle is acute or obtuse. If the angle is acute, use the measure between 0 and 90. If the angle is obtuse, use the measure between 90 and 180. INTEGTE TEHNOLOGY oint out that a graphing calculator may need to be set to record angle measure in degrees, since either degree or radian measure can be selected. This feature is generally used for trigonometry calculations, however. UETIONING TTEGIE What methods can you use to bisect an angle? Which method do you think is the most accurate? Explain. You can use a compass and straightedge, paper folding, or measurement with a protractor. ossible answer: You are more likely to draw the bisector accurately from the vertex by using a compass and straightedge because the method is exact. odule Lesson 2 UIZE THE LEON What is the ngle ddition ostulate and how does it relate to the bisector of an angle? If a ray from the vertex of an angle divides the angle into two parts, the sum of the measures of the parts is equal to the measure of the whole original angle. n angle bisector is a ray that divides an angle into two equal parts. ngle easures and ngle isectors 794

7 EVLUTE Evaluate: Homework and ractice Use a compass and straightedge to construct a copy of each angle Online Homework Hints and Help Extra ractice IGNENT GUIE oncepts and kills Explore onstructing a opy of an ngle Example 1 Naming ngles and arts of an ngle Example 2 easuring ngles Example 3 onstructing an ngle isector ractice Exercises 1 3 Exercises 4 7 Exercises 8 11 Exercises INTEGTE THETIL TIE Focus on easoning.2 iscuss how the rotractor ostulate can be applied in addition to the ngle ddition ostulate to find the measure of angles outlined on top of a protractor. KINETHETI EEIENE Have students work in pairs to write highlighted and prerequisite vocabulary from the lesson on index cards, such as acute, obtuse, and straight angles; angle bisector, and ray. tudents talk about what each term means, then place the cards face down. One student draws a card and acts out the term on the card (for example, a right angle) using hands and arms. The other student guesses. Then they switch roles and the first student guesses while the second student acts out or draws the term. raw an angle with the given name. 4. JWT 5. N J W Name each angle in as many different ways as possible W 1 Z T W, ZW, WZ, and 1 Use a protractor to draw an angle with the given measure G N 2 L L, GLJ, JLG, and 2 odule Lesson 2 Exercise epth of Knowledge (.O.K.) ecall of Information.5 Using Tools ecall of Information.6 recision OON OE athematical ractices J ecall of Information.5 Using Tools kills/oncepts.2 easoning kills/oncepts.5 Using Tools kills/oncepts.4 odeling 22 2 kills/oncepts.4 odeling 795 Lesson 16.2

8 Use a protractor to find the measure of each angle E F VIUL UE emind students to show all arcs and extend segments far enough when creating compass and straightedge constructions. 55 Use a compass and straightedge to construct the bisector of the given angle. heck that the measure of each of the new angles is one-half the measure of the given angle VOI OON EO If students compass settings are not tightly fixed, the compass setting may change without students awareness. tress to students that they must check compass tightness and keep the same fixed compass setting for accuracy. Use the ngle ddition ostulate to find the measure of each angle. 15. m + m = m 40 + m = 70 m = 30 E 16. E m EF + m E = m F 30 + m E = 140 m E = 110 Use a compass and straightedge to copy each angle onto a separate piece of paper. Then use paper folding to construct the angle bisector F odule Lesson 2 Exercise epth of Knowledge (.O.K.) OON OE athematical ractices 23 2 kills/oncepts.4 odeling 24 2 kills/oncepts.6 recision 25 2 kills/oncepts.2 easoning 26 3 trategic Thinking.2 easoning 27 3 trategic Thinking.3 Logic 28 2 kills/oncepts.5 Using Tools 29 3 trategic Thinking.5 Using Tools ngle easures and ngle isectors 796

9 OUNITING TH tudents will have to use algebra together with the ngle ddition ostulate to set up an equation to solve for the variable when one of the angles includes a variable. oint out that angles may not always have whole-number measures. 19. Use a compass and straightedge to construct an angle whose measure is m + m. Use a protractor to check your construction. 20. Find the value of x, given that m = Find the value of y, given that m KL = 135. L (16y) (10x) K N m + m = m x = 112 x = 4 m KLN + m NL = m KL y = 135 y = ulti-tep The figure shows a map of five streets that meet at oncord ircle. The measure of the angle formed by elville oad and Emerson venue is 118. The measure of the angle formed by Emerson venue and Thoreau treet is 134. Hawthorne Lane bisects the angle formed by elville oad and Emerson venue. ickinson rive bisects the angle formed by Emerson venue and Thoreau treet. What is the measure of the angle formed by Hawthorne Lane and ickinson rive? Explain your reasoning. Hawthorne Ln. elville d. oncord ircle Emerson ve. ickinson r. Thoreau t. The measure of the angle formed by elville and Emerson is 1 118, so the measure of the angle formed by Hawthorne and Emerson is (118 ) = The measure of the angle formed by Emerson and Thoreau 1 is 134, so the measure of the angle formed by Emerson and ickinson is (134 ) = 67. y 2 the ngle ddition ostulate, the measure of the angle formed by Hawthorne and ickinson is = 126. odule Lesson 2 IN1_NLEE389762_U716L /19/14 10: Lesson 16.2

10 23. epresent eal-world roblems carpenter is building a rectangular bookcase with diagonal braces across the back, as shown. The carpenter knows that is a right angle and that m is 32 greater than m. Write and solve an equation to find m and m. m + m = m ITIL THINKING eview why students can use the ngle ddition ostulate to find a missing angle measure if they know the measures of one angle and the total angle, in order to find the measure of the other angle, when an angle is divided into two angles that do not overlap. E x + (x + 32) = 90 2x + 32 = 90 x = 29 o, m = 29 and m = = 61 OELING 24. escribe the relationships among the four terms. tudents know the measure of a right angle. iscuss how to use the measure of a right angle to find the measure of a straight angle (180 ) and the total number of degrees in one full rotation (360 ). ngle bisector The definitions of the terms angle bisector and angle are each built upon the definitions of the term below it. The definition of the term ray is built upon the undefined term line below it. efined terms ngle ay 25. etermine whether each of the following pairs of angles have equal measures. elect the correct answer for each lettered part.. KJL and LJ. J and J. LJ and NJ E. KJ and J No Yes No Yes No Yes No Yes No Have students work with a partner to write a guide to copying and bisecting angles in the form of a comic strip. Encourage them to include enough information so that someone who has never done these constructions could follow the procedure. N L J K. JK and J Yes EE TO EE IUION Undefined term Line a. no; m LJ = = 48 m KJL b. yes; m NJ = 48 so m J = = 94 and m J = = 94 c. yes; m NJ = 48 and m LJ = = 48, so m LJ = = 142 and m NJ = = 142 d. no; m JK = 90, but m J = = 94 e. no; m KJ = = 82, but m J = = 94 odule 16 IN1_NLEE389762_U716L Lesson 2 4/19/14 10:33 ngle easures and ngle isectors 798

11 JOUNL Have students name and give the definition of at least five other words that use the prefix bi- to mean two. 26. ake a onjecture rhombus is a quadrilateral with four sides of equal length. Use a compass and straightedge to bisect one of the angles in each of the rhombuses shown. Then use your results to state a conjecture. onstructions may vary. ample: In a rhombus, the bisector of an angle also bisects the opposite angle. H.O.T. Focus on Higher Order Thinking 27. What If? What happens if you perform the steps for constructing an angle bisector when the given angle is a straight angle? oes the construction still work? If so, explain why and show a sample construction. If not, explain why not. Yes; the construction still works. In this case, the construction produces two right angles since each has half the measure of a straight angle (180 ). 28. ritical Thinking Use a compass and straightedge to construct an angle whose measure is m - m. Use a protractor to check your construction. 29. ommunicate athematical Ideas Explain the steps for using a compass and straightedge to construct an angle with 1 the measure of a given angle. Then draw an 4 angle and show the construction. onstruct the bisector of the given angle. Then construct the bisector of one of the angles that was formed. odule Lesson Lesson 16.2

12 Lesson erformance Task store sells custom-made stands for tablet computers. When an order comes in, the customer specifies the angle at which the stand should hold the tablet. Then an employee bends a piece of aluminum to the correct angle to make the stand. The figure shows the templates that the employee uses to make a 60 stand and a 40 stand. UETIONING TTEGIE You are given a 60 angle and an 80 angle. How could you use them to construct a 20 angle? ample answer: opy the 60 angle inside the 80 angle, with the two angles sharing a side. The angle adjacent to the 60 angle will measure = 20. You are given a 70 angle and a 60 angle. How could you use them to construct a 25 angle? ample answer: isect the 70 angle to create two 35 angles. opy a 35 angle inside the 60 angle, with the two angles sharing a side. The angle adjacent to the 35 angles will measure = The store receives an order for a 50 stand. The employee does not have a template for a 50 stand and does not have a protractor. an the employee use the existing templates and a compass and straightedge to make a template for a 50 stand? If so, explain how and show the steps the employee should use. If not, explain why not. Yes; first construct the bisector of the template for the 60 stand to create two 30 angles. Then construct the bisector of the template for the 40 stand to create two 20 angles. Next, copy one of the 30 angles. Finally, copy one of the 20 angles so it shares a side with the 30 angle. The measure of the resulting angle is = 50. You are given a 50 angle and a 40 angle. How could you use them to construct a 5 angle? ample answer: opy the 40 angle inside the 50 angle, with the two angles sharing a side. The angle adjacent to the 40 angle will measure = 10. Then bisect the 10 angle to create two 5 angles. VOI OON EO tudents may have difficulty completing the last step of the Lesson erformance Task, in which they must copy a 20 angle so that it shares the non-horizontal side of the 30 angle. This can happen when students are used to starting with horizontal lines in their constructions. oint out that there is nothing wrong with rotating their papers to start with a horizontal line. odule Lesson 2 ETENION TIVITY The Lesson erformance Task introduces the idea of combining simple constructions to produce more complex ones. Have students think about how they could use this idea to construct a 35 angle from a 30 angle and a 40 angle. Then have them make up problems that apply the idea. Each problem should give the measures of two or more angles and ask how an angle of specified measure could be constructed using the given ones. tudents should provide answers for each of their problems. coring ubric 2 points: tudent correctly solves the problem and explains his/her reasoning. 1 point: tudent shows good understanding of the problem but does not fully solve or explain his/her reasoning. 0 points: tudent does not demonstrate understanding of the problem. ngle easures and ngle isectors 800