DATE PERIOD

-Study Guide and InterventionAngles of Polygong

Sum of Measures of lnterior Angles The segments that connect thenonconsecutive sides of a polygon are called diagonals. Drawing all of the diagonals fromone vertex of an z-gon separates the polygon into n - 2 triangles. The sum of the measuresof the interior angles of the polygon can be found by adding the measures of the interiorangles of those n - 2triangles.

lnterior Angle I lf a convex polygon has n sides, and S is the sum of the measures of its interior angles,SumTheorem I then S= 180(n - 2).

A convex polygon has ffi Tbemeasureofan18 sides. Find the sun of the measures interior angle of aregular polygon is

120. Find the number of sides.The number of sides is z, so the sum of themeasures of the interior angles is 1202.

S: 190(n _ Z)

12An: 180(z - 2)

12On=1802-360-60n : -360

n=6

ffif{?frtrxFind the sum of the measures of the interior angles of each corlvex polygon.

of the interior angles.S=180(z-2)

= 180(13 - 2):190(11): 1980

1. 10-gon

4.8-gon

7. 150

10.165

13. Find r.

2.16-gon

5. 12-gon

8.160

11. 168.?5

3.30-gon

6.3r-gon

Ihe measure of nn interior angle of a regular polygon is given. Find the numberof sides in each polygon.

9. 175

12. 135

(6x+ 10)"

4'.17 Glencoe Geometry

DATE PERIOD

-Study Guide and InterventionParallelograms

Sides and Angles of Parallelograms A quadrilateral withboth pairs of opposite sides parallel is a parallelogram. Here are fourimportant properties of parallelograms.

P-a//s+4

lf POFS is a parallelogram, then

The opposite sides of aparallelogram are cugruent.

ffi=SRandF3=OR

The opposite angles of aparallelogram are congruent.

zP= zRand LS= LQ

The consecutive angles of aparallelogram are supplementary.

tPand LSare supplementary; zSand zRare supplementary;eFland t-Qxa supplementary; z-Qarfr LPare supptementary.

lf a parallelogram has one rightangle, then it has four right angles.

It mLP: 90, then mLQ: 90, mtR: 90, and mtS = 9t).

ffi ffABcDis a parallelogra*, find o and b.

2a: 34

a:17/A and LC arc opposite angles, so tA = LC.8b = LLZb=L4

ffiiluprnFind r and y in each parallelogram.

t. 8y

t-6FlLI8B

/a* p,e/

2.

2y

so,[-ltso72x

4.

6."ffo GlencoelMcGraw-Hill 423 Glencoa Gaometry

DATE PERIOD

-Study Guide and InterventionRectangles

Properties of Rectangles A rectangle is a quadrilateral with fourright angles. Here are the properties of rectangles.

A rectangle has all the properties of a parallelogram.

. Opposite sides are parallel.

. Opposite angles are congfuent.

. Opposite sides are congruent.r Consecutive angles are supplementary.r The diagonals biseet each other.

Also:. All four angles are right angles. LUTSJTSR, LSRU, antd, /RUT are right angles.r The diagonals are congruent. TR = l1g

tffiffi In rectangle RsTaabove, US = 6f, + S and Bf = 7x - 2.Findr.The diagonals ofa rectaagle biseet eachother, so US = RT.

6s+3=7x-Z3:x*25=tc

flHatHfrn

ffi rnrectangleuilTUabove, wLSTR = 8r * S and wLUTR =16r - 9. Find wLSTR.LUTS is a right angle, sowLSTR * wLUTR :90.&+3+16x-9:90

24xt-6:9024x = 96

x=4mLsTR: gr + 3: g(4) + 3or35

ABCD is a reetangle. B

l.lfAE = 36 and CE = 2x - 4,findr.

2.If BE:6y + 2 and CE:4y * 6,findy. A

S.If BC = 24 andAD: 5y - 1, findy.

4.Tf wLBEA: 62, find wLBAC.

5.If m/AED = LZx and rn.LBEC = L0x + 20, find mtAED.

6.1{ BD = 8y - 4 and AC = 7y * 3,frndBD.

l.If ruLDBC : 10r and.rn/ACB : fo;z- 6, find rnLACB.

8. IfAB : 6y and BC :8y, find BD in terms ofy.

9. In rectangLe MNAP,wLL = 40. Find the measure of eachnumbered angle.

o GlencoeAlcGraw-Hill 435 Glencoe Geometry

DATE PERIOD

-Study Guide and InterventionRhombt and Squares

PropertieS of Rhombi e rhombus is a quadrilateral with fourcongruent sides. Opposite sides are congruent, so a rhombus is also aparallelogram and has all of the properties of a parallelogram. Rhombialso have the following properties.

The diagonals are perpendicular. Mfr lHDEach diagonat bisects a pair of opposite angles. Mff bisects IRMO and LRHO.

ffinisects IMRHand LMOH.

ll the diagonals of a parallelogram areperpendicular, then the figure is a rhombus.

ll RHOMiS a parallelogram and RO L MH,

lhen RHOM is a rhombus.

ffi rnrhombusABCD,mSAC = B2.Findthemeasure of each numbered angle.ABCD is a rhombus, so the diagonals are perpendicular and LABE Ais a right triangle. Thus rzZ4 = 90 and m.Lt:90 - 32 or 58. Thediagonals in a rhombus bisect the vertex angles, so mLL = mLZ.Thus, mLZ:58.A rhombus is a parallelogram, so the opposite sides are parallel. IBAC aud l-3 arealternate interior angles for parallel lines, so n'tll = *2.

ffiilffiABCD is a rhombus.

L.If m/ABD = 60, find m.LBDC.

S. IfAB = 26 and BD :20, findAE.

6.If m,LCBD = 58, find m/ACB.

Z.If AE = 8, findAC.

4. Find rnLCEB. A

6.Tf AE : 3.r - 1 and AC : 16, find r.

7.If nTLCDB = 6y and m/ACB :2y + 10, findy.

8. IfA"D = ?at + 4 and CD = 4n * 4, find x.

9. a. What is the midpoint ofE-nZ

b. What is the midpoint ot CJ?

c. What kind of figure is FGHJ? Explain.

d. What is the slope ofF?/?

e. What is the slope of GJ?

f. Based on parts c, d, and e, what kind of {igure is FGIIJ?

441o Glencoe/McGraw-Hill

Explain.

Glencoe Geometry

DATE PERIOD

-Study Guide and Intervention ftontinued)

TrapezoidsMedians of Trapezoids The median of a trapezoid is thesegment that joins the midpoints of the legs. It is parallel to thebases, and its length is one-half the sum of the lengths of the bases.

In trapezoi d, HJI{L, MN : }W"t + LK).

ffi ffi is the median of trapezoid BsrU. Find r.MN=

80=

30=30=5=r

ffiilffi

RS

H3x+5S

+ UT)

-5*9xr*?e.\

3x

L2

ftL(2'

fr6xc

_b)

MN is the median of trapezoid H,IKL. Fiad each indicated value. A---------:!

1. Find MN itHJ= 82 and LK: Go. Uaq

2. Find LKif HJ = 18 and MN : 28.

8. Find MN if HJ + LK: 42.

4. Find wLLMN tf rntLHJ = 116.

5. Find wLJKL it HJKL is isoseeles and wLHLK : 62.

6. Find HJ itMN: 5* + 6,HJ = 3* + 6, and LK - 8x.

7. Find the length of the median of a trapezoid with verticesA(*2,2), B(3, 3), C(7,0), and D{*8, -2).

o Glencoe/McGrawHill M8 Glencae Geometry

Study Guide and InterventionCircles and Circumterence

Pafts of Circlgs A circle consists of all points in a plane that are agiven distance, called the radius, from a given point called the center.

A segment or line can interseet a circle in several ways.

r A segment with endpoints that are the center of the eircle anda point of the circle is a radius.

. A segment with endpoints that lie on the circle is a chord.

. A chord that contains the circle's center is a diameter.

a" Name the circle.The name of the circle is OO.

b. Name radii of the circle.m, Bo,co, and DD are radii.

c. Name chords of the circle.A.B and ffi are chords.

d. Name a diameter of the circle.AA ir a diameter.

ffiffiffin

DATE PERIOD

-

A,a7-',, -eV )P4)

choro:rl EDrat ius: E, E, Ediameter: E

1. Name the circle.

2. Name radii of the circle.

S. Name chords of the circle.

4. Name diameters of the circle.

5. FindAA ifAB is 18 millimeters.

6. Find "4.r8 and AB if -BY is 1^0 inches.

7.IsA.B =ffit Explain.

m

541o GlencoelMcGraw-Hill Glencoe Geometry

PERIOD

-

6F is a rninor arc.

ffi is a major are.

LGEFiS a central angle.

nLHEC+ mLCEF+ mLFEG+ mLGEH:3ffi

ni6F - nzCEF

n;edF-s6o-mOF

nieF+niFG=m€A

{ffiffi rn oft, tnARB = 42 andl? is a diarnetenFiudarffi sllldmffi./ARB is a central angle and, rruIARB = 42, sa nr@ = 42.Thus nffi = 360 * 42 or 318.

ffiAHilM

Study Guide and InterventionAngles and Arcs

Angles and ArCs A central angle is an anglewhose vertex is at the center of a circle and whosesides are radii. A central angle separates a circleinto tvro arcs, a mqior arc and a minor arc.

Here are some properbies of central angles and arcs.r The sum of the measures of the eentral angles of

a cirele with no interior points in common is 360,

. The measure of a minor arc equals the measureof its central angle.

r The measure of a major arc is $60 minus themeasure of the minor are.

r T\vo arcs are congruent if and only if theircorresponding eentral angles are congruent.

. The measure of an arc formed by two adjacentarcs is the sum of the measures of the two arcs.(Arc Addifion Fostulate)

Find each measure.

1. wLSCT

8. ruLSCQ

7, mCD

s. nficD

2, mzSCU

4. wLQCT

T

@lf m/3OA = d,$,, frnd eaeh measune.

6. rnEE 6. lrfid

8. nACB

10.ntfr

o Glencoe/McGraw-Hill 547 Glencoe Geometry

DATE PEHIOD

-Study Guide and Intervention pontinued)

Angles and ArcsAfc Length An arc is part of a circle and its length is a part of the circumference ofthe circle.

ffi hro*, ,L4,,B- 1BE, RB -8randffi is a diameten Find the length atffi.m/ARB: 135, so *ffi = 135. Using the formula C : Zrr,thectrcumference is 2n(8) or 16r. To find the length offfi', write aproportion to compare eaeh part to its whole.

tensthjfffi : degrryqeasuregf 3rg_ proportioncircumference degree measure of circle

( _13516n' 360

" (16nX135)I:-" 360

= 6tt Simplify.

The length ofG is 6zr or about 18.85 units.

ffiffi

$ubstitution

Multiply each side by 1&r.

the diameter of @ is 24 units long. Find the leugthof eaeh are for the given angle rneaslrrer

t.6E if maog: 120

z.frm. if mLDoE * LZo

S. FD tf ruLCOB = 45

*eEi. if mlcoB : 45

T'lre diarneter of OP is l5 units Iong and LSPT a LnPf.Find the length of each are for the given angle meaflre.

6.m if ruLSW : 7o

6. Im tf ruLRW : 5A

7, MST

8.I{ES tf ruLMPS : 140

"ffi,

e Glencoe/Mccraw-Hill 548 Glencae Geomeuy

DATE PEHIOD

-Study Guide and Intervention ftontinued)

Arcs and ChordsDiameters and Chordsr In a circle, if a diameter is perpendicular

to a chord, then it bisects the chord andits arc.

o In a circle or in congruent circles, twochords are congruent ifand only ifthey areequidistant from the center.

Z

tW t44,then 7X = ffi anafu ='frd.ll OX= OY,thenIE=Ett XE = ffi then 7E and ffi are equidistant from point O.

ffi rn oo, 6 tffi,oo= 18, and cD = z/..Findr.A diameter or radius perpendicular to a ehord bisects the chord,so,8D is half of CD.

1

ED :;(24)=L2

Use the P;rthagorean Theorem to find x tn LOED.(ODz + (ED)z = (OD)Z Pythasorean rheorem

x2 + L22 = L52 substitution

x2+144=22512: 81r:9

ffi

Multiply.

Subtrect 144 from eac* side.

Take the square root of each side.

In OP, CD = 24 and ,n& = 45. Find each measure. BC

^@,t,.Aq

4. AB

7.nffi

2. RC

6.nffi8"ntffi

8,QB

6.r?tB

s. nfrD

12, mTS

16. nt&

ln OG, DG = GU and AC * "8f. Find each measure.

TA,TU

18. CD

11. ?E

L4. GD

16. A chord of a circle 20 inches long is 24 inches from the centor ofa circle. Find the length of the radius.

e Glencoe/McGraw-Hill 554 Glencae Geometry

DATE PERIOD -.-

Study Guide and InterventionAreas of Parallelograms

Areas of ParallelOgrams A parallelogram is a quadrilateral with both pairs ofopposite sides parallel. Any side of a parallelogram ean be called a base. Each base has acorresponding altitude, and the length of the altitude is the height of the parallelogram.The area ofa parallelogram is the product ofthe base and the height.

lf a parallelogram has an area of A square units,Area of a Parallelogram I a base of b units, and a height of fi units,

then A = bfi.

ffi Find the area of parallelogram EnGE.A : bh Area of a paraltelogram

: 30(18) b: Bo, h = 18

= 540 Mukipty.

The area is 540 square meters.

ffi

F3'Dl: t4 . lC

r----lo--*-rThe area oI parallelogram

ABCDIS GD. AT.

Find the area of eaeh parallelogram.1trlli /

Find the area of each shaded region.

4.WWZ atd,ABCD arcrectangles.

18ft

3' F-lIf jr'cm

1.6 cm

5. A11 angles are right 6. EFGH and NOPQrectangles; JKLM

areisa

=!EF

7. The area of a parallelogram is 3.36 square feet. The base is 2.8 feet. If the measures ofthe base and height are each doubled, find the area of the resulting parallelogram.

8. A rectangle is 4 meters longer than it is wide. The area of the rectangle is 252 squaremeters. Find the length.

square.

o Glencoe/McGraw-Hill 611 Glencae Geometry

PERIOD

Study Guide and InterventionAreas of Triangles, Trapezoids, and Rhombi

ArgaS Of Triangles the area of a triangle is half the area of a rectangle with the samebase and height as the triangle.

lf a triangle has an area of A square units, a base of b units,

and a corresponding height of h units, ,n"n 4 = |Ofr.

X

ffie: Lrun

= {oaxzal= 336

Find the area of the triaugle.Area of a triangle

b:24, h :28

Multiply.

&"'24m

The area is 336 square meters.

ffiffiffiEn

5r.

Find the area of each frgure.

4F{lh, "***.1

I\\

54I7. The area of a triangle is 72 square inches. If the height is 8 inches, find the length of

the base.

8. A right triangle has a perimeter of 36 meters, a hypotenuse of 15 meters, and a leg ofI meters. Find the area of the triangle.

t

-60-------t

o GlencoellvlcGraw-Hill 617 Glencoe Geometry

DATE PERIOD

-Study Guide and Intervention pontinued)

Areas of Triangles, Trapezoids, and RhombiAreas of Trapezoids and Rhombi ff.e area of a trapezoid is the product of half theheight and the sum of the lengths of the bases. The area of a rhombus is half the product ofthe diagonals.

lf a trapezoid has anb, and b, units, and

e=|n1ar+or).F-br*'l/.ffi

F-&--a

area of A square units, bases ofa height of h units, then

lf a rhombus has an area of A square units anddiagonals of d, and d2 unils, then

e = f,a''a*

{ffi Find the area of the trapezoid.a:f,n{ur+ur)

=-|trrxrs + 4o)

:435The area is 435 square

flmHffiiilre

Area of a trapozold

h-15,4=18,b2=40

Simplify.

meters.

Find the area of eaeh quadrilateral given the cnordinates of the vertiees.

1.

7. The area ofa trapezoid is 144 square inches. Ifthe height is 12 inches, find the length ofthe median.

8. A rhombus has a perimeter of 80 meters and the length of one diagonal is 24 meters.Find the area of the rhombus.

o Glencoe/McGraw-Hill

o

618 Glencoe Geometryr

DATE PERIOD

-Study Guide and Intervention {continued)

Eurface Areas of PrismsSurface Areas of Prisms The surf;ace area of a prism is thelateral area ofthe prism plus the areas ofthe bases.

Suriace Arealf the total surface area of a prism is l"square units, its height is

of a Prismf, units, and each base has an area of I square units and aperimeter of Punits, then I= L + 28.

ffi Find the surface area of the triangular prism above.

Pythagorean Theorem

h2 :27 simptify.

h : 3\/5 Take the square root of each $ide.1

B :; X base x height Areaoratriangle

= ffo)(sV5)or 15.6 cmz

The total area is the lateral area plus the area of the two bases.

? : 180 + 2(15.6) substitution

= 2LL.2 cmz simptify.

rffifffiFind the surface area of each prism. Bouud to the nearest tenth if uecessary.

1.

Fird the lateral area of the prism.

L: Ph Lateral area of a pri$m

= (18X10) p= 18, rr: 10

: 180 cm2 Multiply.

Find the area of each base. Use the P5rthagorean Theorem to find the height of thetriangular base.

1r2 132:62

"rrffi,Am

g.

6.

,rffi-8m

o Glencoe/McGraw-Hill 674 Glencoe Geometry

DATE PERIOD

Study Guide and Intervention ftontinued)

Surtace Areas of CyllndereSurface Areas of Cylind€fs The surface area of a cylinderis the lateral area ofthe cylinder plus the &reas ofthe bases.

Surface Area I lf a cylinder has a surface area of Tsquare units, a height ofof a Cylinder I h units, and the bases have radii of runits, then I= 2qrh + 2rr2.

ffi $ind the sur{ace area of the cylindenFind the lateral area of the cylinder. If the diameter is12 centimeters, then the radius is 6 centimeters.

L:Ph Lateral area of a cylinder: (Znr)h P: zrr:2x(6)04) ,=6,h-14* 527.8 simptifo.

Find the area ofeach base.

B : rrZ Area of a circte: tr(6)g t=6- 113.1 Eimplify.

The total area is the lateral area plus the area of the two bases.T : 527.8 + 113.1 + 113.1 or 754 square centimeters.

WFind the surface area of eaeh cylinden Round to the nearest tenth.

I in.

12 in.

2vd

D.

20 in-

''r**ffi

u'trffi

o GlencoelMcGraw-Hill 680 Glencae Geometry