3. Teorems bou Perpendicular Lines oal Use eorems abou perpendicular lines. Key Words complemenary angles p. 67 perpendicular lines p. 08 eo-civiy Inersecing Lines old a piece of old e paper 3 Unfold e paper. paper o form a line. again by lining up e firs fold. Label e angles as sown. 3 o e fold lines appear perpendicular? Use a proracor o measure eac angle. ow many rig angles are formed? 5 Wrie one or more conjecures abou perpendicular lines. Te eo-civiy above suggess e following eorems abou perpendicular lines. TORMS 3. and 3. Suden elp STUY TIP Teorem 3. Words ll rig angles are congruen. Symbols If ma 90 and ma 90, en a ca. Teorem 3. ells you a if one rig angle is marked on a pair of inersecing lines, en e oer ree angles are also rig angles. Teorem 3. Words If wo lines are perpendicular, en ey inersec o form four rig angles. Symbols If n m, en ma 90, ma 90, ma3 90, and ma 90. 3 n m aper 3 Parallel and Perpendicular Lines
XMPL Perpendicular Lines and Reasoning In e diagram, r s and r. eermine weer enoug informaion is given o conclude a e saemen is rue. xplain your reasoning. r a. a3 ca5 3 u b. a ca5 5 c. a ca3 s Soluion a. Yes, enoug informaion is given. o angles are rig angles. y Teorem 3., ey are congruen. b. Yes, enoug informaion is given. Lines r and are perpendicular. So, by Teorem 3., a is a rig angle. y Teorem 3., all rig angles are congruen. c. No enoug informaion is given o conclude a a ca3. Perpendicular Lines and Reasoning In e diagram, g e and g f. eermine weer enoug informaion is given o conclude a e saemen is rue. xplain.. a6 ca0. a7 ca0 3. a6 ca8. a7 ca 5. a7 ca9 6. a6 ca 6 0 g 7 8 9 e f Suden elp TORMS 3.3 and 3. LOOK K Teorems 3.3 and 3. refer o adjacen angles. or e definiion of adjacen angles, see p. 68. Teorem 3.3 Words If wo lines inersec o form adjacen congruen angles, en e lines are perpendicular. Symbols If a ca, en ^&( ^&(. Teorem 3. Words If wo sides of adjacen acue angles are perpendicular, en e angles are complemenary. Symbols If &( &(, en ma3 ma 90. 3 3. Teorems bou Perpendicular Lines 5
viaion XMPL Use Teorems bou Perpendicular Lines In e elicoper a e rig, are ax and ax rig angles? xplain. Soluion If wo lines inersec o form adjacen congruen angles, as &* and &* do, en e lines are perpendicular (Teorem 3.3). So, &* &*. X ecause &* and &* are perpendicular, ey form four rig angles (Teorem 3.). So, ax and ax are rig angles. LIOPTRS Main roors of a elicoper may ave wo o eig blades. Te blades creae e elicoper s lif power. XMPL 3 Use lgebra wi Perpendicular Lines In e diagram a e rig, &*( &*( and ma 30. ind e value of y. Soluion a and a are adjacen acue angles and &*( &*(. So, a and a are complemenary (Teorem 3.). 6y 30 6y 30 90 ma ma 90 6y 60 Subrac 30 from eac side. y 0 ivide eac side by 6. NSWR Te value of y is 0. Use lgebra wi Perpendicular Lines ind e value of e variable. xplain your reasoning. 7. a ca 8. &( &*( 9. KJ &( KL &(, ajkm camkl 5 36 9y L M z z J K 6 aper 3 Parallel and Perpendicular Lines
3. xercises uided Pracice Vocabulary eck Skill eck. omplee e saemen: If wo lines inersec o form adjacen congruen angles, en e lines are?. Wrie e eorem a jusifies e saemen abou e diagram.. a5 and 6 are 3. j k. ma9 ma0 90 rig angles. 6 l 5 m j 7 8 k g 9 0 f In xercises 5 7, p. Wrie an euaion o find e value of x. (o no solve e euaion.) 5. p 6. p 7. p 5 70 Pracice and pplicaions xra Pracice See p. 679. Perpendicular Lines and Reasoning eermine weer enoug informaion is given o conclude a e saemen is rue. xplain. 8. a ca5 9. a6 ca7 0. a ca3. a ca5 c d 3 c d 5 6 7 Logical Reasoning Wa can you conclude abou a and a using e given informaion?. &( ^&( 3. n m. k omework lp xample : xs. 8 xample : xs. xample 3: xs. 7 n m k 3. Teorems bou Perpendicular Lines 7
rror nalysis Sudens were asked o se up an euaion o find e value of x, given a v w. escribe and correc any errors. 5. 6. v v 56 (x + ) w 9 w (x + ) + 90 = 56 9x = 80 Suden elp I LSSZON.OM OMWORK LP xra elp wi problem solving in xs. 7 is a classzone.com Using lgebra ind e value of x, given a s. 7. s 8. s 9. 60 s 55 0. s. s. (x 0) (3x 5) 0 0x 5x s ngle Measures ind e value of x, given a ^&( ^&(. Ten use e value of x o find ma. 3.. 5. x (x 0) 8x 7x 0x x 6. Window Repair You are fixing a window frame. You fi wo srips of wood ogeer o make e crosspieces. or e glass panes o fi, eac angle formed by e crosspieces mus be a rig angle. o you need o measure all four angles o be sure ey are all rig angles? xplain. You be e Judge 7. In e diagram sown, a and a3 are congruen and complemenary. an you conclude a &*( &*(? xplain your reasoning. 3 8 aper 3 Parallel and Perpendicular Lines
Spors Spors In orieneering, a compass and a map are used o navigae roug a wilderness area. Suppose you are in an orieneering even and you are raveling a 0 eas of magneic nor, as sown below. N 0 W y ORINTRIN Te people sown above are paricipaing in an inernaional orieneering compeiion. S 8. Wa can you conclude, given a NYW and SYW are congruen? xplain. 9. ow many degrees do you need o urn o ravel due eas? 30. ow many degrees do you need o urn o ravel due sou from e posiion sown on e compass? Origami Origami is e Japanese ar of folding pieces of paper ino objecs. Te folds on e paper sown below are e basis for many objecs. On e paper, &* &*. 3. re aj and aj complemenary? xplain your reasoning. 3. If maj maj, wa are eir measures? J 33. Is ere enoug informaion o conclude a aj is a rig angle? xplain your reasoning. Sandardized Tes Pracice Muliple oice In xercises 3 and 35, use e diagram below. 3. Wic of e following is rue if g? ma ma > 80 ma ma < 80 ma ma 80 g None of ese 35. If g and ma 0, wa is ma? 0 50 60 J 0 3. Teorems bou Perpendicular Lines 9
Mixed Review lassifying ngles Sae weer e angle appears o be acue, rig, obuse, or sraig. Ten esimae is measure. (Lesson.6) 36. 37. 38. inding omplemens and Supplemens ind e measure of e angle. (Lesson.3) 39. a is a complemen of a, and ma 37. ind ma. 0. a is a supplemen of a, and ma 56. ind ma. Verical ngles ind e value of x. (Lesson.).. 3. 80 (x 0) 50 (9x 30) (0x 6) x lgebra Skills ecimals valuae. (Skills Review, p. 655). 3.6 9.8 5.. 6. 7. 5.9 7. 79. 9 8. 00.5 6. 9. 30. Quiz Tink of eac segmen on e sopping bag as par of a line. Tere may be more an one correc answer. (Lesson 3.). Name wo lines perpendicular o.. Name a line skew o ^&(. 3. Name a line a appears parallel o ^&(.. Name a line perpendicular o plane. ind e value of e variable, given a p. (Lesson 3.) 5. p 6. p 7. p z 3 57 3 (3y ) 0 aper 3 Parallel and Perpendicular Lines