WARM UP LEARNING GOALS Solve each equation. • Calcu late th e supplement of an angle.

! • Calculate the complement of an angle.

1. 90 = Sx + x • Classify adjacent angles, linear pairs, and vertical angles.

• Use facts about supplementary, complementary,

2. x + 3x = 180 vertical, and adjacent angles and linear pairs in multi­

step problems to write and so lve simple equations for

3. (180 - x) + (90 - x) = 210 unknown angles. Qv'\~ le

4. ~X + X = 180 - X KEY TERMS • straight angle

i • supplementary angles

• complementary angles

• perpendicular • collinear

• adjacent angles

• linear pair

• vertical angles

~---> /7

v~(~~

L. aflste-. ~> l/"1~

You know how to classify individual angles based on their measure. Now consider how

angles are related to each other. What types of specia l relationships exist between pairs

of angles?

LESSON 2: Specia l Delivery • MS-19

Two ang les are

supplementary

angles if t he sum of

t hei r ang le measures

is equa l to 180°.

Two angles are ~ complementary

angles if th e sum of

t hei r ang le measures

is equa l to 90°.

To "d raw"

means

you can

use your measurement

tools . .. so,

get out your

protractor

and

stra ightedge.

In t he prev ious activit y, you created supplementary angles and

complementary angles. l ~ • ~7

Let 's create sets of su pplement ary angles.

1. Use a p10lt acto• te draw a pair of supplementary angles that share a side. What is the measure of each angle?

Ll ..f-L.2..: ts-Q

2. l::l!e e pretl"eetor :to draw a pair of supplementary angles that do not share a side. What is the measure of each angle?

<.t ~ <2-:: l&'O

3. Calculate the measure of an angle that is supplementary to LKJL.

~ J L

ISO 22

MS-22 • TOPIC 1: An g les and Tria ng les

--7 '1.0 Now, let's create sets of comp lementary ang les.

4. ~so a protraeter tE> draw a pair of complementary angles that share a side. What is the measure of e~h angle?

L.. ' +- 4. 2- :: qo

How can you check t hat your

measurements are correct?

5. Yse a prott actor to draw a pair of corryplementary angles that do not share a side. What is the measure of each angle?

"'- ~ ~'~:. 'iO

~~ -6. Calculate the measure of an angle that is complementary

to LJ.

J L_ qo

_(pCJ :,_:;::---;

30

7. Given each statement, write and solve an equation to determine the measure of each angle in the angle pair.

a. Two angles are both co~uent and sueplem~ntary.

x+x = 1a-o 2.. X -::.. l }0 \ x ::: qo · I

b. Two an_gles are both co!:!9Luent and complementary.

c. The supplement of an angle is half the measure of the

angle itself. ---

-----\. s

.aiC Congruent ang les are angles t hat have t he same

measure.

LESSON 2: Special Delivery • MS-23

d. The supple~nt of an angle is 20° more than the measure of the angle itself.

X-\- (X +2.0 \ = \ <60 2.. X+ 20 ::.I ~0

-2... CJ - 2-U (X+2.0) ::. I 00

2.. x :: HoD ____.. -2- '2.-

e. Angles 1 and 2 are com_e lementary. The measure of angle 2 is 10° larger than the measure of angle 1.

x t- (X+to) -:=qo 2.)(--\-lO ?-C[O

-tO -\0

~~~ "2..-- ""2---

f. Angles 1 and 2 are supplementary. The measure of angle 1 is three degrees less than twice the measure of angle 2.

X + ( -z. x -3) -= 1 ~o 3 "X - 3 .:::.. 1 &-0

+3 -t-3 3X::.l,t_3

x::.CDI (2.X-3).::\\t1

~ ~ 8. Use the figure to determine the measure of L ZXY.

w

X y

L Z.X j

MS-24 • TOPIC 1: Ang les and Triang les

Let's explore ang les formed by perpendicular lines.

+---+ ~ 1. Draw a nd label AS ..L CD at point E. How ma ny right angles

are formed?

L1 nght on,les Qre +o{roe.d

+---+ +---+ 2. Draw and label BC .l AS at point B. How many right angles

are formed?

ti n9h \-- ~oj~s ar-<- fo rrne.J

3. Name all angles that you know are right angles in the figure shown. Note: Points A , D, and B lie on the same line segment.

c

8

--1' Two lines, line

segments, or rays are

perpendicular if they

intersect to form 90°

angles. The symbol for

perpendicu lar is .l .

.:y

Compare your

drawings wit h your part ner's

drawings.

What do you notice?

When points lie on

the same line or line

segment, they are said

to be collinear.

LESSON 2: Special Delivery • MS-25

In each of the next three activities you wi ll exp lore special ang le pairs. Let's get started.

WORKED EXAMPLE

L 1 and L 2 are adjacent angles. L3 and L4 are not adjacent ang les. -

YV

1. Use the figures in the Worked .Example to define adjacent angles in your own words. . ()

\w o Qf15\es are. ad.j"aCell+ 1 r +-h~ ho. v (... a c.o mh"' on V u--\-e.')( an J._ a. C.ornMo() S) d.~·

2. Draw and label L2 so that it is adjacent to L1.

Ca n you th ink of other ways to draw L2 so t hat it is adjacent to L1?

LESSON 2: Special Delivery • MS-27

Adjacent angles are

two ang les that share

a common vertex and

share a common side.

How cou ld you illustrate 2 non-adj acent ang les on

patty paper?

3. Is it possible to draw two angles that share a vertex, but do not share a common side? If so, draw an example. Would the angles be adjacent angles? If not, explain your reasoning.

common verk>( no common s .-J~

l)d j a c.e.n + L_

4. Is it possible to draw two angles that share a side, but do not share a vertex? If so, draw an example. If not, explain

your reasoning. I' l S har~ No+ possible.. .. Lr...rwo Qfl~ es

,j LL ShQ(~ {;( t.Jer-k.i' a side., -r-nell -rn~

2. reuj s share. a. t/e/ft.,C

5. Use one sheet of patty paper to create a set of congruent adjacent angles. Describe your process. Label the patty paper "Congruent Adjacent Angles."

6. Use one sheet of patty paper to create a set of non-congruent adjacent angles. D~ibeyour process. Label the patty paper "Adjacent Angles."

7. Are all adjacent angles supplementary? Explain your reasoning.

No ~~ measures of adJ·acenf. t h r(._[arlton

tlflS/(...5 · do no+ ttve.. all:t .f1J ~Ct'-h o+her.

MS-28 • TOPIC 1: Ang les and Tri ang les

8. Are all supplementary angles adjacent? Explain your reasoning.

1\lO 1 s upplemen-1--arj an,j 1-e s not- nece s~aidJ a{jaceJl+.

Let's explore a different angle relationship.

WORKED EXAMPLE

are.

L 1 and L 2 form a linear pair. L 3 and L 4 are do not form a

linear pair.

)I

1. Use the figures in the Worked Example to define a linear pair

of angles in your own words. . f' ~wo tlr1jles are a JJ/->eor f.aJT' ' .rh ~ are a J J ilcen + llfl d -rf, e, ('

n Ofl ~ comma n J i d e_J PurrTJ Ill. G(le .

LESSON 2: Specia l Delive ry • MS-29

- - --- 11 A linear pair of ang les

is fo rmed by two

adjacent ang les that

have noncommon

sides that form a line.

2. Draw L2 so that it forms a linear pair with L1 II '= .gf att . . · -sQ O!:IQ SH9et P .r ~aper 811d I ecVI d your r espo,se. baeel yeur patt '

pap9r Lmear Pair." ~ L 1 + L 2. - t f?O •

,3j Name all linear pairs in the figure shown.

L I a!?d L 1

L I t?JncJ L 3 L 2. ~lld L '-/

L '2. and L 3

4. If the angles that form a linear pair are congruent what you conclude? , can

1 {}--the. an3les ~a+ PorM a tinetJtr peul' art. (!,()IIJrUt/1-f

1 -fhe/1 -fh~ tnkrJee--h//j iJnes.)

line. seJmen-1-.s' or- ra., }Jiles a r~ per pell dt"eular.

5. What is the difference between a linear pair of angles and supplementary angles that share a common side?

--rhere... ,·.5 no J,f'FerefJce. A /,near f~l' of anJies f'orm a /,i?eJ a11d Jo do tt f ().;/ o f S" f f I ef71.-(,-/l -far :J ti/lj /e J -fl.c. -f S har~ a c.otnm. 11'/1 .s,· de...

MS-30 • TOPIC 1: Ang les and Triangles

6. What is the difference between a linear pair of angles and supplementary angles that do not share a common side?

-rhe- pfM·r of Juff.Je.mea.far'J a nJie s Qr<! not adj oct11 +, buf a l,i'Jear ft:ul"

tnuJ+ be. aJ.Jace11+·

7. Angle ABC and angle CBD form a linear pair. Write and solve an equation to determine the measure of LABC.

A

)(f3~::t~O -~} -3}

)( -::::. l '1 2-

8 0

8. The angles shown are a linear pair of angles. Solve for x.

• 4x~ •

9. Write and complete the sentence on your Linear Pairs patty paper.

If two angles form a linear pair, then the sum of the measures

of the linear pair of angles is ---f/J!IJ--=-=----· , ~Q •

LESSON 2: Specia l Delivery • MS-31

Is there another way

you could draw L2 so t hat it form s a

vertica l angle pair wit h L 1?

Let's explore one more special angle relationship.

WORKED EXAMPLE

L 1 and L 2 are vertical angles. L3 and L4 are not vertical ang les.

1. Use the figures in the Worked Example to define vertical

angles in your own words. d . f V er-hc·a I a f13 1 e S are.. nar> a ~a Cl'll

on j I e. s far rtJ~ d lo j 1-w O

l n -kf' s~c.-h11j hl'l es .

2. Draw L2 so that it forms a vertical angle pair with L 1.

MS-32 • TOPIC 1: Ang les and Tri ang les

3. Name all vertical angle pairs in the diagram shown.

L I Q(ld L ~

L.. 2 and L ~ 4. Trace the figure in Q sti1 n on two diffe sneets of

patty paper. e-- ure you n~bertne-~mgles . Use the patty paper nvestigate ~easures of vertical angles. What do you notice?

Sam.e. SJt.e­

tbnjruen +

5. Use yo• 1r protraetor to measure each an-gle in Question 3.

¥

What do you notice? (l CJ p rcHrtJCfV

Vtr..-h~ l Gflj /e~ are mu.J1- tnow bJ CAJt1j rU til-t ' . ,p · L

t!Lf-'J()' ~ d/J.

6. Use what you know t supplem~ngles and linear pairs to justi your investigationSin Questions 4 and 5.

.S tt f f I.L tne/l kr~ Ollj leJ Uflear pcur- OflJI.u arc..

anJ 114ffU tnL/l~~Y!f

7. Label one sheet of your patty paper "Vertical Angles" and write and complete the sentence.

When two lines intersect to form vertical angles, each pair of vertical angles _ ______ _

have. ~ ~a me.. me4Sure.

Vertical angles are

two nonadjacent

angles that are formed

by two intersect ing

lines.

How ca n you rotate t he patt y pa per t o invest igate ve rt ica l a ngles?

LESSO N 2: Speci al Delivery • MS-33

8. Write and sol four a I . ve an equation to d ng es m each d" etermine the 1agram. measures of all

ver.hc4 I Ur'lj~S

X -cs:: so 2. ")(-qo:: &O

a.

V u-h c t.l ofl5 Je5L

Qfe. C.Ofl ~ ruerr

~ b. -.share. a s;.k - S"ffle rnRJI-fcttj

MS-34 • T OPIC 1· A . ng les and Triang les

-5 ::.

X +S.X::: '~o Co X :::: I ~6 -- --Co (o

X.: ?;o

TALK the TALK

Extra Special Delivery

Answer each question. Draw a figure to justify your response.

1. Two intersecting lines form how many

a. pairs of supplementary angles? .. ~ l Qt\.d ~ ...

L-\ t? ~i r.S L '1. 4t\ ~ L ;,

r ", 4/1.~ L '-/

4. 'f 41'l L I b. pairs of complementary angles?

do no+ FuvfY'I CAmrJer mef>+arj ()(15 l!-~

c. pairs of adjacent angles? , . Ll 01\.o l"L

t..l (-' Q..l r s ~ 'l- ~,. ~ '- ~ ,~ CM.l.LY

"" c.t a.n.!. " I d. linear pairs of angles?

,(_ l Qt'\-~ .t. '2.

t..t o~rs J,1.. "w~ L ~ r L. 11 4~~ "·A

J.. "l 41\'! ' \

e. pairs of vertical angles? . Ll on-~ L. ~ z t?().JrJ •1 r L "1. c.,.~ ~ "'1

2. Suppose two lines intersect. If you are given the measure of one angle, can you determine the measures of the remaining angles without using a protractor? Explain your reasoning.

jC5, 1P :JIVe() -rh~ measure.. of' on~ Onjle ..f'hell ..yhe. ;n24Jur~ of CitJJ~ -lhc;f IV/ In

L ftlJr 1 , +he.. .some . -1h e ~w o

avu~~ . . /'. 17j J~s ~ac-h fi;/fY'I ~ 4/J~4r fur r e ma .. ll11 /) '1 "" "L'.

\M 1-Yf, ~ j 11/t:ll m t!a,j Lit'<- 1 .f'U b..frac.-1- rTdfYJ

Lro ~ ca11 deWmill~ -rhe. me~J't-tr~ tY+' )

-cac.,h r~ma...rui1,J t:tnsJu.

LESSON 2: Speci al Delivery • MS-35

\

\

~ \

~ ~ ~\

~ ~ ~

·a. Describe the relationship between vertical angles.

VerJ.tca Ls hav~ ...rh~ .same YneA.JU(e...

b. Describe the relationship between adjacent angles.

UcL ja C€11+ QnJ}eJ Qre..,

J4ppJ.emM .fctrJ

:\ 4. Draw and label a diagram that includes at least one of each relationship. Then identify the angles that satisfy \

each description. ~ ';"/ <. l .f.(. 2. :::. qo

CD r ~/' l • complementary angles supplementary angles perpendicular lines adjacent angles linear pair vertical angles

MS-36 • TOPIC 1: Angles and Tri ang les

~>