Points, Lines, and Rays
Obj&cII". kletltily andlabel points, ~r>&S.
liIlII segments,andrays.
Learn About It
Ttle widest cable-stayed bridge in the world spans the Charles River in Boston. In the picture you can identify many geometric features.
L9<IrnlrdP.Zakim &nke'NiHBtidge
~ Apoint is an e~acllocatiO!1 in space. , • Read: point C Write: C
~ A lineis anendlessstraightpathmade Read: line COo, line DC
up01 a continuouscollection01 points. , , Write; CDorDC
~ A line segment is a parI of Read: line segment CD
a line arld hastwo endpoints. , , or line segment DC
- -
Write: CD or DC
~ A ray has one endpoint and extends Read: ray CD
wiltlout end in one direction. , , Write: cD (The endpoint
is always the first leller.)
~ A plane is a collection of points , Read: plane JKL
S;;, , ':z
that Iorms a tlat. con~nuous, and (The:3 ~ners can bein
unellding surface. , , any order.)
~ Intersecting lines have one point
::?Z Read; line ASintersects
in common, line COat point E.
, .
... Perpendicular lines intersect atright angles.
II" Parallel lines lie in Ille same plane anddo nol intersect.
390
~ perpendicularRead: Line RTtoisline WX.
- -
Write: RT.1. WX
" N Read: Line MNis parallel
• • • 10 line PO.
p 0 Write: ;;:mUPO
Guided Practice Name each ligure.
1. T ,. , , •
• •
3. oS , , •• , M
~~Yourse/f
• Whk;I>lene" 00 ILJSe to nameeactlligu'''?
• 00 IneOOto writaIha
",na,s in a carta", oroar?
Explain Your Thinking "'Why doCOand DCname different rays?
Practice and Problem Solving Name each ligure.
•. , :---: , 6.~" ~ ,. ããc,---, u ,
Describe each pair 01 lines. Us,e symbols II possible .
•. ,
~ ~ o
••
Draw and label each llgure.
11. DR 12. OF J.. LN 13. CO 115X 14. plane EBT
Solve.
1$. Four paraliellines are perpendicular to two otherlines. Athowmany pointsdo the lines intersect?
Write About It E!lplain how
perpendicular lines are one special case of intersecting lines.
Mixed Review and Test Prep Open Response
For each exercise, write the Iracllons thai are equivalent. Then circle tile Iracllon In Ille pair thaI Is In simples! lorm.p 9. L.-.nlll
18. Which of these-a line, a line segment. or a ray-can contain the other two?
18. How many lines can intersect at a point? Use a diagram to explain your thinking.
Multiple ChoIce
23. Identify this figure. Pl. 15,~ II
N M
I
, , ,
10. 'ij. 4' 12
. , ,
21. 18'5'2
, . ,
20. 'ij.16ã 16
, , ,
22. 5' J' 15
A line MN B point MN
c plane MN Dray MN Chapler 15 Lesson 1 391
Work Together
HandsOn
Measure, Draw, and Classify Angles
ObJeclive Measure, draw, andClassifyaogl&$.
Vocabularydegree. I
rlghleogle acule engle obtu.e ang~
elr.lghl angle Angles are formed by two rays with a common endpoinl The
common endpoint of the rays is called the ve"e~of the angle.
A small arc is used to identify the inside. or interior, of an angle.
Materials
protrac1orOf
leam;ng Tool 12 To name an angle. you can nante three
points on the angle-a point on each ray ami the venex in the middle. You can also name an angle just by naming its verte~ if no other angles share thaI venex.
The symbol L is used to identify an angle.
A protractor is a1001 used to measure angles in degrees (').
Follow the steps below to measure LFDEand LBDe
,
LMXH. LHXM, LX
Measuring Angles
"T~ Place Ihe cenler mark of the
o protraCtor on the vene~. D. Align the 0" mark of one of the protractOf scales wilh ona ray of the angle,
• ,
392
.'" o Find where Ihe olher ray passes
Ihrough the same scale. Read the measure 01 the angle on that scale .
• What is Ihe measure 01 LFDE?
LBDe?
• How can you tell when to use the inside scale and when to use Ihe oulside scale 01 the protractor?
"
You can also use a prolractor 10 draw an angle of a gWen measure.
Draw an angle that measures 75'.
Drawing Angles
"UA> On a sheet of paper. draw and
o label a ray.
, I
Place the center mark of the protractor on the endpoint 01 the ray. Align the ray with the 0' mark of one ot the protractor scales. The endp6nt of the ray will be the vertex of the angle.
Using the scale on which the ray aligns with (1'.mark the p6nt at 75'.
Label the p6nt.
,,"fEA> Draw a ray from the vertex throvgh
o the p6nt you labeled. Write the name of the angle .
• What is fhe measure of the angle?
'Which p6nt is the vertex of the angle?
,
"
... ~'..
.,,~ ., ..
.~ ...
',' .. '
" .'
" .,
., .,
- -.\-'.
Chapter 15 Lesson 2 393
You can classify an angle by its measure.
Classifying Angles
The measure ot a right angle is equal to 90'.
rightLJKL
The measure ot an obtuse angle is greater than 90"and less than t8O".
The meaSUfe Of an acute angle is greater than 0' and less than 90".
The meaSUfe of a straight angle is equal to 180".
ob1 ....LCDE
On Your Own
,
IUIlghtLxrz
In Exercises 1-4, use symbols to name each angle three diHerent ways.
,.
, ,
$. Which angle has a greater measure. LeGRor LZVP?
Classlly each angle as acute, obtuse. straight. or rIght.
.. V o ,
,
394
' j
"/\ .oJ
Use a protractor to draw an angle having each measure.
Classify each angle as right. acute, obtuse, or straight.
14. 165' 18. 50"
IS. 90"
10.115'
16, 20"
20. 10"
17. 65"
21. 135'
You have learned how to classify. draw, and measure angles.
22. Is the sum of the measures of two acute angles always less than 90"? Explain why or wtly not
23. How could you lisllhe kinds of angles you know-right straight, acute. alld obtuse-in order from least to greatest measure? Explain.
Visual Thinking
Construct Perpendicular Lines
Vou can use a compass and a straightedge to conSlruet perpendicular lines.
Follow these steps to construct perpendicular lines.
~'b Draw linec and point Was shown al the
o right. Pul the compass point on W. Draw an arc that inlersects line c atlwO points.
Label these pointsX and Y.
.w
"
lines can1be named uS<"Il lowercase Iehel'$.
.U... Place the point of the compass atX and draw
o an arc below line c. From point Y. use the same compass measure and draw an arc below line c. Label the intersection point V.
"
, • w ,
flo'£'" Draw line wvand label itd. Line d is
e perpendicular to line c. w
, x ,
, ,
Chapter 15 Lesson 2 395
Hands On
Triangles
Objective ClaWlytriar>gles and lindnUsing W>gIe meaSUreS.
Learn About It
A triangle is made up of 3 line segments called sides. Each pair of sides has a common endpoint, or vertex. and 10nns an angle.
.. You can classify triangtes by the lengths 01 their sides.
Vocabulary
eoqulI~a."
Iso_I..
~.~
Materials
straighlacJga
equilateral triangle All sides are the
same length.
Isosceles triangle At leastlWO sides are the same lellQlh.
scalene triangle No sides are the
same length.
III' You can also classify triangles by their angle measures.
~right triangle~.. ~.
one light angle
acute triangle all acute angles
obtuse triangle one obtuse angle
Try this activity with a partner to learn about angle measures in a triangle.
'!o"'~ Use a stra>ghtedge todraw a triangle.
o Cut rt oul.label the angles a. b, and c.
'!oU", Tear off the three angles a1the triangle.
o
396
",TE",
o Arrange the angles to make a straight angle.
• What Is the measure 01 a straight angle?
• What Is the sum of the angle measures in a triangle?
• Does this work fOl any triangle? Explain.
Guided Practice
Classlly each triangle In two ways. Then lind the missing angle measures.
~:s\l YOurself
• Aret....anglesacuta?
obi,,",,? righl?
• Whatis1hIlsum01 I....angIE> maawres ina trianglE>?
Explain Your Thinking'" Is an equilateral triangle also an isosceles triangle?
bplain wily or why not
Practice and Problem Solving
Classlly each triangle Intwo ways.
3. 2m
,vm . ... "" ~. ....
X Algebra. Expressions Wrlle an expression to represent a.
Then lind the value 01 a.
, A ~
a ~ ~ • Ai
n "A ~
Use s straightedge and protractor lor Problems 11 and 12.
1 I. What's Wrong? Ari says that an 12. Try to draw each of the following. If a isosceles triangle can also beobtuse. Is figure cannot be drawn. explain why.
Ari rigllt or wrong? Draw triangles to
help you explain, • a scalene acute triangle
13. Create and Solve Use the sum of the angle measures in a triangle to write and solve yom own triangle problem,
• an equilateral righttriallQle
• a scalene right triangle
E.<traPrilCllceSee woe419.Sete, Chapter lS Lesson 3 397 20. Find p. bplain how you
found your answer.
(Ch. 15, ~31
Mixed Review and Test Prep
14. ~x~ ~x.! , ,
". . , ". '6 x 10
~xj , , , .
". " 25 X S ". '8 x 11
rOpen Response
Multiply. Write each product In simplest lorm. p. 12. t..oon21
Congruence
Objective ldentily~ liguresand oongo'uen1parts01 figur&s_
Learn About It
Rgures thai are the same sire and shape are called congruent figures. The symbol ill is used to
indicate congruence. Corresponding parts of congruent figures are congruent
Which ligures in the photograph appear to be congruelll?
I"I""I'".,...,...I'!/' Vocabulary
Different Ways to Check for Congruence
Way 0 You ClIn use 1"'1elng.
IIyou Irace triangleABCand place tile tracing On lop 011,Ian91e DEF. you will lind thallhe triangles a'o congruent.
,
C 8 F ~
MBC b.DEF
So, AS DE, Be EF, CA FD.
Also, LA LD, LB LE. and LC LF.
Way 0 You can use a ruler and a protractor.
In an equilateral triangle. the three sides are congruent arid thtt three angles are congruent. Small lines indicate congruent sides aOO congruent angles.
Another Example
Squares
These squares are nol congruent. They have the same shape, but they are not the same size.
398
'''---1--'''',
JK KL JL LJ LK LL
00
Guided Practice
Trace eacllllgure. Mark tile congruent sides and tile congruent angles.
'j /
M\t. YoUrself
• Whichsidesara ilia
~-,
• Whichangles Mva tI>asarna measure?
Explain Your Thinking" Draw three squares on a piece of paper. Can you divide each square differently into lour congruent pans? Show your work.
Practice and Problem Solving
Trace each figure. Use a ruler to measure the sides and a protractor to measure tile angtes of eacllfigure. Mark tile congruent sides and angtes.
']-1: ,. wr---.'
z'---~, :z lR
Use Ille diagram to answer tile quesllons. Explain your reasoning.
6. What is the length of DE?
7. What is the measure of LA?
6. What is the measure of LF?
O. What is the length ofDF?
10. What is the measure of LD?
•
Open Response
Find each statistic for the data below.
p,.s.l-...2)
90.75.80.80.80
MUltiple Choice
15. Which of the following statements
:~c:tS::'ALB_LD cEF_BC
11. mean 13. mode
12. range
14. median - -
B AC_ DF D LA_LD ExIraPrilC1lCeSee Il30e4t9.SetC, Chapter 15 Lesson 4 399
Quadrilaterals and Other Polygons
Obj&cliv. kl9ntily.classify,and ~9po!yQon5.
Learn About It
A quadrilateral is a four-sided figure. The sum of lhe angle measures in any quadrilateral is 360".
In a city you will see many things that are like quadrilaterals.
There are many different kinds of quadrilaterals.
You can use sides and angles to classify them.
Classifying Quadrilaterals
Vo(abui.la~r:;Y'~_i
quadrlll'teôll
"" .. ""
.~ul polygon
' '
quadrilateral foor skies lour angles
L
r
rectangle
opposite sides congruem four right aogles
square
four congruent sides lour right angles
r , , o
parallelogram opposite sides coogruent
and parallel
rhombus four congruent sides opposite sides parallel
trapeZOidU
onty one pair of parallel sides
A quadrilateral is one type 01 polygon. A polygon is a closed figure that has three Of mom sides. Each side is a line segment. 3Jld the sides meet only althei. endpoints.
Polygons Not Polygons
400
A regular polygon is a polygon with all sides congruent and all angles congruent.
Polygons
Name hamples ..~ EKampies
Triangle
D 6 OCUgon 0 U
, .... 8 sides
Ouadrilat....al
D D '00",",
0 0
4sides 9 sides
~nlagon
0 Q Decagon 0 (3
, .... 10sides
HeKagOfl
0 D Undecagon 0 0
, .... 11 sides
Heptagon
0 [] ""'"""-
0 ~
7sides 12sides
A diagonal of a polygon is a segment that joins two vertices of a polygon but is oot a side.
- -
ADand BEare \w() diagoJlals of this he~agon.
E ,
Chapler 15 Lesson 5 401
Classify eaeh polygon in as many ways liS yo... ean.
Guided Practice
1...J L
, r '0 '0
Mk loutJe/f
• How many sides dollst!l<t
porygon "&ve?
• Ate any sides parallel?
• Are any sides congNOOt?
• Are anyanglesC(Ir\QnIeflI'
Explain Y01.Ir Thinkingill'Use the drawing at the right to explain why the s... m of the angle measures in a quadrilateral is 360".
~ I
' - ' .
Practice and Problem Solving
Classify eaeh polygon In as many ways as you ean.
'0 . I I '0
Write polygon or not8 polygOfl to elassify eaeh lig... re.
II possible,lind the messure of each missing angle.
122'
H.
". Ow K
ããD ,. ,~.
".
Solve.
lS. Draw a quadrilaternllhat is not a parallelogram and has two pairs of congruent sides.
la. Is f'Nery square a rhombus? Is every rhombus a square? Ellplain.
H. Draw several parallelograms. including special cases-squares. rectangles. and rhombi. For each figure draw the diagonals. In which kind of parallelogram are the diagonals perpendicular? congruent?
402 Extra Practice seepage 4t9.Set0
leo' ~ 1
180" x • 180" ~ • 190' ~ •
180" x •
~.
•
"'.
•
, ,
Nunmo. ot Int..io<
T<I.ngl..
•
• , ,
""- .... ..
2. Write a formula for finding the sum of the angle measures in any polygon.
3. Useyour formula to find the number of degrees in an OCtagO<l, Complete the table below.Then answer these questions.
1. What pattern do you see between the number of sides of tile polygon and the number of interior triangles?
'+ '0 'v
Math Reasoning Sum It Up
The sum of the angle measures of a triangle is 180ã. You can use that fact to find the sum of the angle measures of any polygon.
Classify each Ilgure In as many ways liS possible. ....,... '-21 Check your understanding 01 Lessona 1-5.
Find the missing angle meaaurea.lLM*>M 3-<11
' A V
'n V 'lJ '~k
10. Afe the figures in Exercises 2 and 8 congruent?
Explain how you know. '--"'51
Chapter lS Lesson 5 403
HandsOn ' " -.- rawt'10Usloo ....I,l""'...
Rotations, Reflections, and Translations
Objective I<lentily andmodelTraMIaIions. rotaTions.
and,ell&ctions.
Vocabulary
tran.1Ofmatlon .eflecTlon .ollltion lran.laflon
Work Together
A Iransformatlon changes the position. but IlOtthe shape. of a plane figure. Renections. rotations, and translations are three kinds of
transformations.
• r 1
refledlon figure flips ove' a line
rolatlon
figure turns about a point
Iranalallon figure slides a given distance in a givan direction You can describe a rotation using the 360" of a circle.
• A
,..
a quane, tum Clockwise about point A
",.
a half tum counterclockwise
about point A Work with a partner to model transformations.
",.
a three-quaner tum clockwise about poin, A
404
.'" o
.'" o
Use a ruler to draw a right trlangle on grid paper. Shade arid cut out the ,nangle.
Ou,line the cut-ou' trlangle on a new sheet of grid paper. Label the tnangle with A Then draw arid label point 0 on the grid paper as shown.
A
+
.'" o
Rotate triangle A a ha~ tum
coonterclockwisa about point O. Outline the triangle. Label the triangle with B.
• What uansformation did you perform?
• How many degrees did you rotale the triangle?
Nowrefle<:tlriangle 8 across a Vel1ical line through point O. Outline the triangle
Label1tle triangle withC.
• What transformation d;d you perform?
Rotate triangle Ca ha~ tum clockwise about point0 asshown. Outline the triangle. Label the triangle withD.
• Is triangle 0 congruent to triangleA?
Usa a transformation to find out.
• Show ano1tler way tousa re!le<:tions.
rotations, Of translations to trans!Orl11 triangleA into triangle D.
o •
8 •,
o 0 • 8 ,
On Your Own
Tell whelher each figure shows a translation, refledlon, or rolatlon.
If a figure shows a rotallon. name file number of degrees of rOlallon .
••
H-.L ~
. -. I '
Copy each figure onto grid paper.
Tllen complete the given transformatlon.
"mtranslation
••
rotation 01 90' clockwise reflecllOrl
Chapter 15 Lesson 6 405
On grid paper, copy trlangte A. Label point O. Draw and label the figure In each new position for EKerelses 7-11.
7. Translate triangle A 2 units 10 lhe right Labellhe new lriangle 8.
8. Translate triangle A 5 units down. Label the new triangle C.
II. ROlate l1iangle A 90ã counterclockwise about point 0.
Labellhe new lriangle D.
10. Reflect lriangle D across a vertical line through point 0.
Labellhe new lriangle E
11. Whal one transformalion can be used 10 move lriangle A 10 the position shown by triangle C?
12. Which picture shows a reflection of lhe shaded figure?
o
+- +-
~
t +-
~
~
•• •• ••
13. Which picture shows a rOlalion of the shaded figure?
•• •• •• ••
14. Which picture shows a translation of the shaded figure?
•• •• •• ••
15. Donya says lhat reflecling a righl triangle across its base is lhe same as rotating it 180"' aboul its righl angle. Is she rig hI? Draw a diagram 10 eKplain.
406
16. Create and Solve Draw a design on grid paper. Wrile steps to change the design using rOlalions. refleelions. and lranslations. Draw the solution.
You learned how to Identify and model rellectlona, rOlatlons, and tran$latlon$.
17. Explain how rellectlons and rotations are alike and different.
Use words or a diagram.
18. Explain how rot3lions. reflections. and translations can help you decide jf two fjgures are congruent.
How to Play
Tangrams
If you don't have tang,ams or Learning Tool 52. you can make your own pie<:es by tracing the ones on this page and cUlling them out.
You may want 10 make a sketch of your design to help you remember it if your panner can'l solve lhe puzzle.
The pieces are removed and mixed up. The second player must decide how to arrange the p1eces within the outline, wilhoul any
ove~appingp1eces.
Take turns repeating Sleps::> and 3.
The first player makes a shape with the tangram pieces and traces its outline. The p1eces should notove~ap. At leaSI some 01 them should beplaced edge 10 edge, 2 players
What You'll Need • Leaming TOOl 52 Ofa sal 01 tang<am piec(ls like tl>8 ones shown,
o o
o o
Chapter 15 Lesson 6 407
• Use pattern blocks to make a model 01 Vi's pattern. Trace the panern and cut it out.
• Yo!J know that a translation moves a figure a given distance
in a g;ven direction. So you can '::=====:;-~==---7---:J-,
translate the pattern to begin r
the tessellation.
• You know that you can rotate figures 180".So use rotation to fill in the gaps.
Solution; Vi's pattern tessellates.
You can make a model to help you solve Ihe problem.
ThiS is what you know;
• A tessellatioo or tiling is a repeating panern that covers a plane without gaps or overlaps.
• There are four trapezoids in the pattern.
Look blJ(;k at the problem. C/ln you use/I different strategy to check the answer?
• _ _ 21111JS1en0I'Id1Ii"",laoid
Problem-Solving Strategy
Make a Model
Problem In social stlldies class, Vi and her classmates have been S1lldyin9 tessellation patterns from a dome on a bllilding. A tessellation is a repeating pattern that covers a plane without gaps or overlaps. Vi made a tile using foor trapezoid pattern blocks. Will Vi's pattern tessellate?
408
Guided Practice
Unthe Mk Yourte" quUtions10 help you soIye uchproblem.
1. Hamid cut asmaU square 'rom one side of I large square and translated it to theother side 01 the square. WII Hamilf. pellem tesselate?
.---JI I
I I
2. Jan ctJI OUI two pentagons.Only one
pentagon tessellates. Which one i$it?
@ To tessellate. the sum of the angles that meflt mustbe360".
Independent Practice
Milke II model toa.oIVe e.c:h problem.
3. Brittanydesigned this panem. Will it tessellate? Explain whyorwhy IlOl
~sk Yourself
.m'. WhalIKISdotknow?
~ DId I rnodet?
Dldl ~to_
It' tlw pMIemIllIlogeltwf'?
• DId INpUtthe patMm ~
...lcouId_lIl1~
• DidI tilltthe . , . wlu-oI ;..-
~ . .~?
~ Did I"""" tlw po:tllm?
4. What's Wrong? Ricky $lIid that a regularoctagon will tessellate. Is he right orwrong? Howdo you know?
$. c ....tto end $o1Ye Startwith arectangle. Create I pattern that will tessellate. Then create a dilferent pattern that wi.
IlOl tessellate. Tntde patterns with a classmate. Then tel ~
wIlictl of the two pallems will lessellate. ...
Chapter 15 lesson 7 409
Mixed Problem Solving
SOlVI. Show you. _k. Tell.mat strategy you used.
.. A panel of 5 ardlitects sit inII row. Juan istotheright or Mike.. MaYis istothe IefIof rlra. Mike ison Yulu"sright. Juan isOIl oneend. Name!heir order from leftto r'ighL