Solve each equation. EXAMPLE. Name &1 in two other ways. &AEC and &CEA are other names for &1. Quick Check

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1 -6. Plan Objectives o find the measures of angles o identify special angle pairs xamples Naming ngles Measuring and lassifying ngles Using the ngle ddition Postulate Identifying ngle Pairs 5 Making onclusions rom a iagram -6 What You ll Learn o find the measures of angles o identify special angle pairs... nd Why o find the measures of angles in flower arrangements, as in xercise. Measuring ngles heck Skills You ll Need O for Help Skills Handbook page 758 Solve each equation.. 50 a m n x z y New Vocabulary angle acute angle right angle obtuse angle straight angle congruent angles vertical angles complementary angles supplementary angles inding ngle Measures Math ackground ngle measure and segment measure have important similarities. ongruent angles can be moved onto one another so they match exactly, as with congruent segments. ongruent angles are also indicated with tick marks. Like a ruler, the intervals on a protractor must be equal. he most common unit of angle measure is the degree, which results when a circle is divided into 60 equal parts. inally, the ngle ddition Postulate corresponds directly with the Segment ddition Postulate. More Math ackground: p. Vocabulary ip You may also refer to the angle suggested by the two segments and Q as &Q. n angle (& is formed by two rays with the same endpoint. he rays are the sides of the angle. he endpoint is the vertex of the angle. he sides of the angle shown here are and Q. he vertex is. You could name this angle &, &Q, &Q, or &. MPL Naming ngles Name & in two other ways. Q Lesson Planning and Resources See p. for a list of the resources that support this lesson. ell Ringer Practice heck Skills You ll Need or intervention, direct students to: Solving Linear quations Skills Handbook, p. 758 Quick heck 6 hapter ools of eometry Special Needs L Students with fine motor difficulties may have problems using a protractor. You may want to have students work in pairs so they can assist each other as needed. & and & are other names for &. a. Name & two other ways. l, l b. ritical hinking Would it be correct to name any of the angles &? xplain. No; ' have for a vertex, so you need more info. in the name to distinguish them from one another. One way to measure an angle is in degrees. o indicate the size or degree measure of an angle, write a lowercase m in front of the angle symbol. he degree measure of angle is 80. You show this 80 by writing m& = 80. elow Level L emonstrate on an overhead projector how to line up a protractor with a ray, and how to choose the appropriate scale. Have students choose the scale for which one of the sides passes through zero. 6 learning style: verbal learning style: visual

2 Key oncepts Postulate -7 Protractor Postulate Let O and O be opposite rays in a plane. O, O, and all the rays with endpoint O that can be drawn on one * side of can be paired with the real numbers from 0 to 80 so that a. O is paired with 0 and O is paired with 80. b. If O is paired with x and O is paired with y, then m&o = ux - yu. nline You can classify angles according to their measures. acute angle 0 < x < 90 Note the special symbol that s tucked into the corner of the right angle. When you see it, you know that the measure of the angle is 90. MPL Measuring and lassifying ngles ind the measure of each angle. lassify each as acute, right, obtuse, or straight. a. b. right angle x 90 obtuse angle 90 < x < 80 straight angle x 80. each uided Instruction rror Prevention iscuss as a class why it is inappropriate to name & as &. sk: How could this cause confusion? here are three angles whose vertex is. xplain also that the measure of an angle does not need a degree symbol. Math ip sk: How is the Protractor Postulate like the Ruler Postulate in Lesson -5? oth pair numbers in a one-to-one correspondence with geometric objects and use absolute value to determine measurements. uditory Learners Have students take turns with a partner explaining the Protractor Postulate in their own words. MPL MPL onnection to lgebra Review the meaning of the inequality symbol. Visit: PHSchool.com Web ode: aue-0775 Quick heck 0, obtuse 90, right ind the measure of each angle. lassify each as acute, right, obtuse, or straight. a. 0; acute b. 90; right c. 0; obtuse dditional xamples Name the angle below in four ways. ngles with the same measure are congruent angles. In other words, if m& = m&, then & > &. You can use these statements interchangeably. ngles can be marked alike to show that they are congruent, as in this photograph of the ir orce hunderbirds precision flying team. Lesson -6 Measuring ngles 7 &, &, &, & ind the measure of each angle. lassify each as acute, right, obtuse, or straight. m& 0, obtuse; m& 80, acute dvanced Learners L Have students write their first names using block letters. hen have them count the number of acute, right, obtuse, and congruent angles suggested by the letters. learning style: visual nglish Language Learners LL Show how the concept and notation for congruent angles is closely related to congruent segments. or example, if, then =. Likewise, if & &, then m& = m&. learning style: visual 7

3 MPL Visual Learners raw the figures for the ngle ddition Postulate on the board. Have students place other points in the interior of &O to reinforce the concept of interior. dditional xamples Suppose that m& = and m& = 88. ind m&. m& 6 uided Instruction eaching ip fter students read the definition of vertical angles, ask: What is another way to define vertical angles? opposite angles formed by two intersecting lines rror Prevention! Students sometimes confuse complementary and supplementary angles. One ways to keep them straight is to remember that c comes before s in the alphabet, just as 90 comes before 80.. Quick heck he ngle ddition Postulate is similar to the Segment ddition Postulate. MPL Identifying ngle Pairs Key oncepts Postulate -8 ngle ddition Postulate Helvetica ondensed imes Roman urostile xtended Markerelt hin In each font, a capital suggests vertical angles. If point is in the interior of &O, If &O is a straight angle, then then m&o + m&o = m&o. m&o + m&o = 80. O O What is m&sw if m&rs = 50 and m&rsw = 5? m&rs + m&sw = m&rsw Using the ngle ddition Postulate 50 + m&sw = 5 Substitute. m&sw = 75 If m& = 5, find m&. 5 Some angle pairs that have special names. vertical angles two angles whose sides are opposite rays complementary angles 50 0 ngle ddition Postulate Subtract 50 from each side. adjacent angles two coplanar angles with a common side, a common vertex, and no common interior points supplementary angles two angles whose measures two angles whose measures have sum 90 have sum 80 ach angle is called the ach angle is called the complement of the other. supplement of the other. R S W hapter ools of eometry 8

4 Quick heck MPL Identifying ngle Pairs In the diagram identify pairs of numbered angles that are related as follows: a. complementary & and & b. supplementary & and &5; & and & c. vertical 5 & and &5 a. Name two pairs of adjacent angles in the photo below. nswers may vary. b. If m& = 7, find m&. 5 Sample: l and l; l and l onnection to Language rts Students are familiar with the word compliment. Point out that the word in this lesson has an e instead of an i. sk students to find non mathematical contexts where complement and supplement are used. MPL dditional xamples When entering the roadway, turn and look for oncoming traffic regardless of what you see in the rear-view mirror. 5a. Yes; the congruent segments are marked. b. No; there are no markings. c. No; there are no markings. d. No; there are no markings. Whether you draw a diagram or use a given diagram, you can make some conclusions directly from the diagrams. You can conclude that angles are adjacent angles adjacent supplementary angles vertical angles Unless there are marks that give this information, you cannot assume angles or segments are congruent an angle is a right angle lines are parallel or perpendicular Name all pairs of angles in the diagram that are a. vertical l and l; l and l b. supplementary l and l; l and l; l and l; l and l c. complementary none 5 Use the diagram from xample. Which of the following can you conclude: & is a right angle, & and &5 are adjacent, & &5? l and l5 are adjacent. nline Visit: PHSchool.com Web ode: aue-0775 Quick heck 5 MPL Making onclusions rom a iagram What can you conclude from the information in the diagram? & > &, by the markings. & and &, for example, are adjacent angles. & and &5, for example, are adjacent supplementary angles, or m& + m&5 = 80 by the ngle ddition Postulate. & and &, for example, are vertical angles. 5 an you make each conclusion from the information in the diagram? xplain. a. W > WV b. PW > WQ a d. See left. c. V ' PQ d. V bisects PQ. e. W is the midpoint of V.Yes; the congruent P W Q segments are marked. V 5 Resources aily Notetaking uide -6 L aily Notetaking uide -6 dapted Instruction L losure sk: How are angles classified? y their angle measure: acute (R 90, right ( 90, obtuse (S 90, and straight ( 80 What are the special angle pairs? vertical, adjacent, complementary, and supplementary Lesson -6 Measuring ngles 9 9

5 . Practice ssignment uide -, hallenge 8-9 est Prep 50-5 Mixed Review Homework Quick heck o check students understanding of key skills and concepts, go over xercises, 8,,, 7. actile Learners xercises 9 Using a corner of paper to model a 90 angle makes classifying acute and obtuse angles visually apparent. rror Prevention! xercise 6 Point out to students that two conditions must be met in this exercise. RISS or more exercises, see xtra Skill, Word Problem, and Proof Practice. Practice and Problem Solving O Practice by xample for Help xample (page 6 xample (page 7 Name each angle in three ways... lyz, lzy, ly M P Y Z lmp, lpm, l, or l Use the figure at the right. Name the indicated angle in two different ways.. & l, l. & l, l raw and label a figure to fit each description See margin. 5. an obtuse angle, &RS 6. an acute acute, & 7. a straight angle, & 8. a right angle, &HI Use a protractor. Measure and classify each angle. about the angle formed by the skis. 60; acute 90; right. 5; obtuse PS uided Problem Solving L nrichment Reteaching dapted Practice L L L Practice Name lass ate Practice -6 raph each point in the coordinate plane.. (, 5. (5,. (0, 6. (, 0 5. (, ind the distance between the points to the nearest tenth. 6. L(,, M(, 7. N(, 0, P(, 8 8. Q(0, 0, R(0, 9. S(0, 5, (0, 0. U(, 0, V(, 0. W(, 7, (, ind the coordinates of the midpoint of each segment. he coordinates of the endpoints are given.. (6, 7, (,. (, 5, (,. (,, (7, 8 5. O(0, 0, ( 5, 6. H(.8,., I(., J(, -, K(, - 8. he midpoint of is (,. he coordinates of are (, 6. ind the coordinates of. 9. he midpoint of is (,.he coordinates of are (,. ind the coordinates of. 0. he midpoint of is (, 7. he coordinates of are (, 0. ind the coordinates of.. raph the points (,, (, 5, (, 5, and (,. raw the segments connecting,,, and in order. re the lengths of the sides of the same? xplain.. crow flies to a point that is mile east and 0 miles south of its starting point. How far does the crow fly? Quadrilateral PQSR has coordinates as follows: P(0, 0, Q(,, R(8,, and S(7, 6.. raph quadrilateral PQSR.. What is the perimeter of PQSR? he oordinate Plane L Pearson ducation, Inc. ll rights reserved. xample (page 8 xample (page 9 0 hapter ools of eometry. ind m& if m& = 5. ind m&j if m& = and m& = 79. Name an angle or angles in the diagram described by each of the following. 5. supplementary to &O lo or lo 6. adjacent and congruent to &O lo 7. supplementary to &O lo 60 O 8. complementary to &O lo or lo 9. a pair of vertical angles lo and lo or lo and lo In the diagram above, find the measure of each of the following angles. 0. &O 90. &O 0. &O 50. &O 0 J 5. What is the midpoint of QR? 0

6 xample 5 (page 9 6. Yes; you can conclude that the angles are adjacent and supplementary from the diagram. 8. Yes; you can conclude that angles are supplementary from the diagram. pply Your Skills. Yes; you can conclude that ' are vertical from the diagram. Real-World onnection Japanese flower arranging makes precise use of angles to create a mood. 7c. nswers may vary. Sample: he sum of the l measures should be 80. x an you make each conclusion from the information in the diagram? xplain.. &J > & Yes; the markings show they are congruent. 5. &J > & 6. &J and & are adjacent and supplementary. See left. 7. m&j = m& J 8. m&j + m& = 80 See left. 9. J > 0. is the midpoint of J. Yes; there are markings.. & and &J are vertical angles.. bisects &J. See left. In the diagram, ml 65. ind each of the following.. m& 5. m& 65 stimation stimate the measure of the angle formed by the hands of a clock at each time. 5. 6: : :00 0 xercises 8. : : :0 80. lower rranging In Japanese flower arranging, you match a stem that is vertical with 0. You match other stems with numbers from 0 to 90, in both directions from the vertical. What numbers would the flowers shown be paired with on a standard protractor? 5, 75, and 65, or 5, 05, and 5 lgebra Use the diagram, below right, for xercises 5. Solve for x. ind the angle measures to check your work. 5. See margin.. m&o = 7x -, m&o = x + 8, m&o = x +. m&o = x -, m&o = 5x + 0, m&o = x +. m&o = 8, m&o = x -, m&o = 6x 5. m&o = x +, m&o = 7x, m&o = 6x - 6. Multiple hoice If m&mqv = 90, which expression can you use to find m&vqp? m/mqp m/mqv m/mqp m/vqp M O Q V N P xercises, 5 sk: Why are three letterers used to name the angles in xercise 5 but only one letter is used in xercise? Vertex J and vertex each apply to only one angle, but many angles share vertex. rror Prevention! xercise 6 Students may be misled or confused because the drawing is not drawn to scale with m&mqv = 90. Students can redraw the figure, but by examining the answer choices students should be able to identify the correct answer.. ; mlo 8, mlo ; mlo 50. 8; mlo 0, mlo 50; mlo 0. 8; mlo 8, ml O 5; mlo ; mlo, mlo 9; mlo O nline Homework Help Visit: PHSchool.com Web ode: aua-006 x 7. a. lgebra Solve for x if m&rqs = x + PS and m&qs = 6x b. What is m&rqs? m&qs? ; 7 c. Show how you can check your answer. See left. R Q S lesson quiz, PHSchool.com, Web ode: aua-006 Lesson -6 Measuring ngles 5 8. rawings may vary. Samples are given. 5. R S H I

7 . ssess & Reteach hallenge 8. bisects &, bisects &, bisects &, bisects &, bisects &, and H bisects &. If m& = 6, find m&h. 0 Lesson Quiz Use the figure below for xercises. 9. echnology Leon constructed an angle. hen he constructed a ray from the vertex of the angle to a point in the interior of the angle. He measured all the angles formed. hen he moved the interior ray. What postulate do the two pictures support? ngle dd. Post.. Name & two different ways. &, &. Measure and classify &, &, and &. 90, right; 0, acute; 0, obtuse Use the figure below for xercises.. Name a pair of supplementary angles. Samples: l and l, l and l. an you conclude that there are vertical angles in the diagram? xplain. No; no angle pairs are formed by opposite rays. lternative ssessment Have students draw diagrams to illustrate the ngle ddition Postulate. hen have them write examples that use each postulate to find a missing measurement when two of the three measurements are known. est Prep Resources or additional practice with a variety of test item formats: Standardized est Prep, p. 75 est-aking Strategies, p.70 est-aking Strategies with ransparencies O est Prep Multiple hoice Short Response Mixed Review for Help Lesson -5 Lesson - Lesson wo angles are congruent, adjacent, and supplementary. What is the measure of each? cannot be determined 5. wo angles are congruent and complementary. What is the measure of each? H. 80 J. cannot be determined 5. wo angles are adjacent and supplementary. What is the measure of each? cannot be determined 5. When 5 is subtracted from the measure of an angle, the result is the measure of a right angle. What is the measure of the original angle? H H. 05 J You are given that m& + m& = m&. a. raw a diagram to show the above. a b. See margin. b. If m& = and & is obtuse, what are the least and greatest whole number measures possible for &? xplain. Use the figure at the right for xercises If 5 75 and 5 8, what is? If 5 9, 5 x, and 5 x, find x. hen find and. x = 8; = 9; = Writing xplain the difference between an orthographic drawing and an isometric drawing. See back of book. ind one counterexample to show that each conjecture is false. 58. he quotient of two integers is 59. n even number cannot have not an integer. 5 as a factor. 0 = 5, so 0 has 5 as a 5 hapter ools of eometry factor and 0 is even. 5. [] a. b. n obtuse l measures between 90 and 80 degrees; the least and greatest whole number values are 9 and 79 degrees. Part of l is. So the least and greatest l measures for l are 79 and [] one part correct