You used slopes to

identifu parallel and

perpendicular lines.I I rliri:li ii i ... i:

t=TennesseeGurriculum Standards

/ 3108.4.8 Apply propedies

and theorems about angles

assoclated with parallei andperpendicular lines to solve

problems.

r' 31.08.4.21 Use properties

of and theorems aboutparallel lines, perpendicular

lines, and angles to prove

ilasrc theorems in Euclideangeomelry.

@ Recognize angle pairs

ffi that occur with parallel

lines.

ffi Prove that two lines

ffiirare parallel.

When you see a roller coaster track, the

two sides of the track are always the same

distance apart, even though the track

curves and turns. The tracks are carefully

constructed to be parallel at all points so. ,

that the car is secure'on the track.

$ ldentify Farallel Lines The two sides of the track of a roller coaster are parallel,ffi and all of the supports along the track are also parallel. Each of the angles formed

between the track and the supports are corresponding angles. We have learned thatcorresponding angles are congruent when lines are parallel. The converse of thisrelationship is also true.

F**€*Easts *.4 Converse 0f Corresponding Angles Postulate .trl

lf two lines are cut by a transversal s0 that corresponding angles are

congruent, then the lines are parallel.

Examples ll 11 = 13, l2 = 14,, l5 = 17,16 = 18, then d ll ,.

The Converse of the Corresponding Angles Postulate can be used to constructparallel lines.

#*e*,es€sas*€E**e Parallel Line Through a Point Not on the Line

E{?ld Use a,strgightedge to

draw AB. Draw apoint Cthat is not on+ €AB.Draw CA.

Copy lCABsolhaI C :

is the verlex of the ',

new angle. Label the i

intersection points D I

andE i

EM Draw CD.Because l

IECD= lCABbvconstruction and they

are corresponding

angles,.(__--+ <-+AB ll cD.

l------,--Euclid's Postulates The

father of modern geometry,

Euclid (c. 300 a.c.) realized

that only a few postulates

were needed to prove

the theorems in his day.

Postulate 3.5 is one ofEuclid's five originalpostulates. Postulate 2.1

and Theorem 2.'10 also reflecltwo of Euclid's postulates.

The construction establishes that there is at least one line through C that is parallel toTB.The following postulate guarantees that this line is the only one.

8:1s.t*:E+tE.c"+;# i$.S Parallel Postulate

lf given a line and a point not on the line, then there exists exacflyone line through the point that is parallel to the given line.

P@

Parallel lines that are cut by a transversal create several pairs of congruent angles. Thesespecial angle pairs can also be used to prove that a pair of lines are parallel.

3.5

3.6

EE.se'fryer"ri* Proving Lines Parallel

Alternate Exterior Angles Converself two lines in a plane are cut by a transversal so I

that a pair of alternate exterior angles is congruent,then the two lines are parallel.

i

Consecutive lnterior Angles Converself two lines in a plane are cut by a transversalso that a pair of consecutive interior angles issupplementary, then the lines are parallel.

. nmt4+mt5=tB0,then/llq

3.7 Alternate Interior Angles Gonverself two lines in a plane are cut by a transversal sothat a pair of alternate interior angles is congruent,then the lines are parallel.

3.8 Perpendicular Transver.rl Conurrrtln a plane, if two lines are perpendicular to thesame line, then they are parallel.

Itp Lr andqtr,rhenpll(

l

"iUeatiry, farafl,et I Lines r, . ;,,'r :',',r:r:i,;ii

Given the following information, is it possible to provethat any of the lines shown are parallel? If so, state thepostulate or theorem that justifies your answer.

a. l'1, = 1611 and 16 are alternate exterior angles of lines (. and n.

Since l7 = 16, (. ll nby the Converse of the AlternateExterior Angles Theorem.

b. 12= 13

12 and l3 are alternate interior angles of lines (. and m.

Since l2 = 13, (. ll mby the Converse of the Alternate Interior Angles Theorem.

;, m

il'RmJ 206 Lesson 3-5 j Proving Lines paraltet

*u!dedFraeti*e

14. 12= l8lC. 172= 174

1E. ml8 + ml73:180

18.

1D.

1F,

l3 = 117

l1 = 115

l8= 16

0PEN ENDED Find wIMRQ so that a ll 6.

Show your work.

Read the Test ltem 6

From the figure, you know that wIMRQ : 5x -l 7 and wIRPN - 7x - 21. You areasked to find the measure of IMRQ.

Solve the Test ltem

IMRQ and IRPN are alternate interior angles. For lines a and 6 to be parallel,alternate interior angles must be congruent, so IMRQ = IRPN. By the definitionof congruence,mlMRQ: mlRPN. Substitute the given angle measures into thisequation and solve for r.

wIMRQ : wIRPN *.ltsriiai* i+[tur*r **gl**

5x + 7 : 7x - 27 G-*=ti9Ectt+*

7 : 2x - 27 S**ire*t 5.s fre= ****: *idr.

28 : 2x &*d *; :* s*ffft $t*+.

14 : x *ieiil* s*+* *i** *y ?.

Now, use the value of x to find IMRQ.

wIMRQ:5x*7 3*{:*tllxtim

: 5(l4l + 7 ;..= .t4

: 77 *Fr++li{g,

CHECI{ Check your answer by using the value of r to ftnd mlRPN.

mlRP :7x - 2L

:7(141 - 21. or 77 {Since %IMRQ : mlRPN, IMRQ = IRPN and a ll 6. {

G$idedFraetice

2. Find y so that e llf. show your work.

Angle relationships can be used to solve problems involving unknown values.

$tudyTipFinding What ls Asked For

Be sure to reread testquestions carefully to be sureyou are answering thequestion that was asked. ln

Example 2, a common error

would be to stop after you

have found the value of xandsay that the solution of theproblem is 14.

Sludg?ipProving Lines Parallel When

two parallel lines are cut by

a transversal, the angle pairs

formed are either congruent

or supplementary. When a

pair of lines forms angles

that do not meet thiscrlterion, the lines cannotpossibly be parallel.

Prove Lines Parallel fne angle pair relationships formed by a transversal can beused to prove that two lines are parallel.

HOME FURNISHINGS In the ladder shown, each rung isperpendicular to the two rails. Is it possible to provethat the two rails are parallel and that all of the rungsare parallel? lf so, explain how. If not, explain why not.

Since both rails are perpendicular to each rung, the railsare parallel by the Perpendicular tansversal Converse.Since any pair of rungs is perpendicular to the rails, theyare also parallel.

3. R0WING In order to move in a straight line withmaximum efficiency, rower's oars should beparallel. Refer to the photo at the right. Is itpossible to prove that any of the oars areparallel? If so, explain how. If not, explainwhy not.

Given the following information, determine which lines,if any, are parallel. State the postulate or theorem thatjustifies your answer.

1.

@

5, SHORT RESPONSE

Find r so that m ll n. Show your work.

6. PR00F Copy and complete the proof of Theorem 3.5.

Given: l1 = l2Prcve: (.ll mProof:

l1=13l3 = 110

2. 12= l54. ml6 + ml\ = 780

a.11=12b. l2= 13

c. 11=13d.?

a. Given

b.?c. Transitive Property

d.?

208 I Lesson 3-5 | Proving Lines Parallel

7. RECREATI0N Is it possible to prove that the backrestand footrest of the lounging beach chair are parallel?If so, explain how. If not, explain why not.

Given the following information, determine whichlines, if any, are parallel. State the postulate ortheorem that justifies your answel

8. 17= 12

10. l5= l712. ml3 + ml6:78014. l3= l7

L 12= 19

11. ml7 + ml8 :180

13. l3= l515. 14= l5

17.16.

Find x so that m ll n.Identify the postulate or theorem you used.

18'*

22. FRAMING Wooden picture frames are often constructed using a miter box or miter saw.These tools allow you to cut at an angle of a given size. If each of the four pieces offraming material is cut at a 45o angle, will the sides of the frame be parallel? Explainyour reasoning.

23, PR00F Copy and complete thq proof of Theorem 3.6.

Given: 17 and 12 are supplementary.

Prove: (.llm

Proof:

ir :rl;l +: f ;.',:

f + ii!

(2x + 25)'

(6x - 144)"

a,

b. l2 and l3 form a linear pair,

a, Given

b.?c,?d.?e,?

c,?d. 11=13e. t.ll m

24. CRAFTS jacqui is making a stained glass piece. She cuts the top andbottom pieces at a 30' angle. If the corners are right angles, explainhow Jacqui knows that each pair of opposite sides are parallel.

PR00F Write a two-column proof for each of the following.

25, Given: l7 = l3ACIIBD

Prove: AB ll CD

26. Given: WXIIYZ,/1 = /2

Prove: WV tt XZ

DC

28.27.

A

Given: IABC = IADC

mlA -t ryIABC:Prove: AElco

A

W

Given: 11 = l2

U ttttProve: XU t ttttf

JKI

1

180

M

MAIIBOXES Mail slots are used to make the organizationand distribution of mail easier. In the mail slots shown,each slot is perpendicular to each of the sides. Explainwhy you can conclude that the slots are parallel.

30, PR00F Write a paragraph proof of Theorem 3.8.

PR00F Write a two-column proof of Theorem 3.7.

STAIBS Based upon the information given in the photoof the staircase at the right, what is the relationshipbetween each step? Explain your answer.

Determine whether lines / and s are parallel. Justify your answel

31.

32.

33.

ffiit:ffi} 210 : Lesson 3-5 ] Proving Lines Parallel

34. 35.

36. # MUITIPLE REPRESENTATI0I,IS In this problem, you will explore the shortest distancebetween two parallel lines.

a' Geometric Draw three sets of parallel lines frand (., s andt, and4andy. For each set,draw the shortest segment BC and label points A and D as shown below.

BAT;CD

b' Tabular Copy the table below, measure IABC and IBCD, and complete the table.

c' Verbal Make a conjecture about the angle the shortest segment forms with bothparallel lines.

37. EHR0R ANAIYSIS Sumi and Daniela are determining which lines areparallel in the figure at the right. Sumi says that since l7 = 12,WllXZ. Danieladisagreesand says thatsince 11 = 12,WXllYZ.Is either of them correct? Explain.

38. REAS0NING Is Theorem 3.8 still true if the two lines are not coplanar?Draw a figure to justify your answer.

39, CHALLENGE Use the figure at the right to prove that two linesparallel to a third line are parallel to each other.

40, 0PEN EN0E0 Draw a triangle ABC.

?. Construct the line parallel to BC through point A.

b. Use measurement to justify that the line you constructed

C. Use mathematics to justify this construction.

41. CHATLENGE Refer to the figure at the right.

a. If ml| -f ml2 - 180, prove that a ll c.

b, Given that a ll c, if mlL 1- ml3 - 180, prove that t L c.

is parallel to BC.

42, WnlTlNG lN MATH Summarize the five methods usedin this lesson to prove that two lines are parallel.

$.W WR|TING IN MATH Describe a situation in which two parallel lines are cut by atransversal and a pair of consecutive interior angles are both supplementary andcongruent. Explain.

W

6

c

2

z

211

sPt 3108,4,9

44. Which of the following facts would be sufficientto prove that line /js parallel to XZ?

I

A l7= l3 C l7= lZB l3= lZ D 12= lX

45, ALGEBRA The expressiont/52 + t/717 isequivalent to

F13c 5!/i5

What is the approximate surface area ofthe figure?

5 in.

4.5 in.I in.

C 202.5 in2

D 2161n2

46.

H 6\rcI 1s\fr5

A 101.3 in2

B 108 in2

F4G16H58

47. SAT/ACT If x2 :25 andU2 :9,what is thegreatest possible value of (, - y)2?

164K70

Write an equation

48. m:2.5, (0,0.5)

in slope-intercept form of the line having the given slope and y-intercept. li-:r:i:i:t; .: i

49. m: [, {0, -o) 50. m: -t,F,-Z)

51, R0AD TRIP Anne is driving 400 miles to visit Niagara Falls. She manages to travel thefirst 100 miles of her trip in two hours. If she continues at this rate, how long will ittake her to drive the remaining distance? :Li:;:ni; j -:;

Find a counterexample to show that each conjecture is false. r|+-:*i-rr 'j-l i

52. Given: 17 and 12 are complementary angles.

Conjecture: l7 and 12 forrn a right angle.

53, Given: points W, X, Y, and Z

Conjecture: W, X, Y, and Z are noncollinear.

Find the perimeter or circumference and area of each figure. Round to the nearest tenth. jli::,Sijt ;'il55. 3.2 n54. I

/qin. ,

i,,1.1 m:

---1

it

56, ,,, i,.,.

.i;,i '"...

.,i.-".. .--.1*-"] * . --i:'6cm

57. Find x and.y so that BE and AD are perpendicular. irt ijcii !-il

212 I Lesson 3-5

(12y -10)"

I eroving Lines Parallel