Common Core Geometry Regents Review:

Topic 1: Proofs

Definitions and Theorems:

Definitioris:

1. 1 lines form right angles

2. angle bisector dvide an angle into 2 congruent angles

3. Complementary angles are two angles whose sum = right angles

4. A midpoint divides a segment into two congruent segments

5. Two angles that form a linear pair are supplementary

G. A bisector divides segments into two congruent segements

Postulates:

1. Reflexive- AB = AB or < A = <A

2. Transitive- I f A B - B C or if <1 = < 2

BC = CD < 2 - < 3

Then AB = CD then < 1 ^ < 3

3. Addition- congruent segments + congruent segment are congruent

congruent angles + congruent angles are congruent

4. Subtraction - congruent segments - congruent segments are congruent

Congruent angles - congruent angles are congruent

Theorems:

1 . Right angles are congruent

2. Straight angles are congruent

3. Complements of the same angle are congruent

4. Supplements of the same angles are congruent

5. Complements of congruent angles are congruent

6. Supplements of congruent angles are congruent

7. Vertical angles are congruent

Examples:

j i . Given: Paralleiogram A B C D with diagonal AC drawn

A D

B C

Piove: A A B C - ACDA

2. In the diagram below of cirx-le O, tangent EC is drawn to diameter AC- Chord BC is paraile:

tu secant ADE. and chord AB is drawn.

P: i'o\; BC C'A EC

AB

3, isoscfif,s trapezoid ABCD liils b;ises DC and ilB witli nonparallel logs AD and BC. St;gnionts

AE. BE. CE, and DE are tirawn in trapezoid ABCD siicli timt ACDE ADCE, AE 1 DE, ami

Pro\r' AA.DE ABCE and prove AAEB is an isosceles triangle.

KelK' is completing a jaroof based on the figure i.Kdow.

She was given tJiat A\Which p a i r o l cx>rrcs would not prove I

\ A ( X D F a n d S A S

C2} EC - EF aiKl SAS

AEDF, and hm ahvadv proven AB DE ing paiTs and t.riangk^ temgniencv suctijsa

^ AxDEF r

o]) /C-Ts Z F a m I AAS

/4j ACBA - AFEDmulASA

I n quadriiareral BLUE shown l>flo\¥, BE ^ UL,

Z i n the diagnnn below, i f A A B E - A C D F and AEFC is dmwix then it could be proven that qnadrilateral ABCD \s a

(3) reclangle

(4) parallelogram (1) square

(2) riiombus

Prove; AG//\ A T N A

Given: Circlt: O, cliorcis AB and CD int(;rsect at £

Theorem: i f two chords interrect io a circle, the product of the lengths of the segments of one chord is equal to the product of the leiiglhs of the segments of the otl-ier chord.

Prove this theorem hv nrovin£ AE - EB =CE-'ED.

10' Given: Parallelognim A B C D , EFG, and diagonal DFB

A B

Prove: A D E E - ABGF

f Given: Quadrilateral ABCD s>Ath diagonAs AC and BD that bisect each other, and Z l = Z 2

C

A D

Prove: A.ACD is an isosceles triangle and A A E B is a riglst triangle

Topic 2: 50H, CAH, TOA, Law of Sines, Law of Cosines, and Area

Formulas Law of Sines

ct _ h _ c

sin A sin B sinC

Law of Cosines

c' = cr -lahcosC

Area of Triangle

k = —nhsinC 2

50 H CAH TOA

s i n ^ X c o s ^ X t a n ^ X H H A

Examples:

CL Given: Eight triangle ABC with r igl i t angle at G

M sill A increases, does cos B increase or tlecrease? Explain u'hy.

Z . Boll places au lS-f<H)t larldef 6 fVet from the bast: tif Ills liou-sr- and leans it up agiunst the side of his liouse. Find, to tlie nearest degree, the measun: of the angle the Iwttom of the ladder makes uith the grmtiid.

3 A man was parasailiiig above a lake at an angle of elevation of 32' f rom abiiat , as mo<I<4efi in the diagram below.

129.5 m

.5 meters ot cable coiiiiectetl the boat to tlie parasail, oximatelv bow inanv meters alxjve the lake was the man?

(1) 68.6

(2) 80.9

E3) 109.8

f4)

As shown in the diagram Ix-lfHv. an island (/i is due north of a marina (M). A boat lioiLse (//i is 4.5 miles due west of the tmuina. From the lx)at house, the island is located at an ;mgle of 54''' fn)m the iTiariina.

I

H 4.5 mi M

Determine and state, to the nraresl tenth of a mile, the di.stance i'rom the htynt house iH) to llie island (/).

DetermiiK' and state, to tfie nearest tenth of a mile, the distanee from the islarni f / - to tlie inarimi • .\/},

h' A ladder 20 toct long leans against a bi i i idiug, foriniug an angle oi 71° with the level ground. To the nearesi Joot, how l i ig i i up the wall of t tu ' bui ldnig does the ladder touch the bnihling?

CD 15 ;3i 18

(2) 16 E4) 19

). Freda, xvho Ls training to nsv a radar system, tietects an aiiplane ilviug at a constant speed and heaiftny in a straight line to pass (lircctlv aver her locMtion, She sees the ahpiane al .m angle oi elevation of 15 and notes that it is maintaining a constant altitude of 62511 feet. Que niinnte later, she sees the airplane at iin angle of ehnaition of 52''"'. !!f>w far ha.s the aiiplane traveled, to the nmresi foot?

Determine and state the speed of the airplane, to the nearest mile- per hour.

When instincEcd to find the length of Hj in right triangle HjG, Aiex wmte the eqiiation

2H^ ~ 7^ wliiie Mailene WTote cos 62^ = Are both students' etjtiations currect':^ sm

Ixplain whv.

/O- I B A / \ B C , where A C is a right aiigW, cos A = . Wliat is sin

o (3) I

' O

C2) V 2 i

In the diagram lielow, a window of a house is 15 feet above tlie grmmd. A ladder is placxai against the house with its base at an angle of 75' with the gi'onnd. Determine and state the length of the ladder t(s the nearest tenth of a fooL

/£- As iiifxieied below, a movie is proiecied onto a large outdoor screen, Tiie bottom of the 6(J-foot-ta!l screen is 12 feet off the ground. The pnqeetor sits on the ground at a luuizontal distance ot 75 feet from the screen.

60

12

75

Determine and .state, to tfie nearest tenth of a degree, the measure of 0, the projection angle.

Z -- As shov^ni in die diagmiii below, tbe angle of elevation f rom a on the ground to tbe top o f the tree is i:M°.

I f the point is 20 feet from tfie base of tlie tree, what Is tbe height o f tbe tree, to the nearest tenth of a fool?

( I ) 29.7 (3) 13.5

i2) 16.6 (4) U . 2

I f Tlie diagram below S1UA\-S a ram[) coiuiecTing tbe ground to a loading pUttlonn -1.5 k-ci above the ground. The ramp measures 11.75 feet, from the ground to the top of tbe ItjiKiiiig platform.

Detenui]ie and state, to the nmrest degree, the angle of ele\n formed bv the ramp and the ground.

lb' Which expres.sioo is always eqmvaletit to siu x when 0° < .r < 9 C ?

{ . ! ) C O S (90° - x) (3) cos (1%)

(2) cos ( 4 5 ' ^ - x ) (4) cosx

r>. As sliowu in the diagram below, a ship is heasling direelly toward a ligiuhouse whose Ix'aeon is 125 feet ahm-e sea level At the first sigiiHng, point A, the angle ofeltnution from tbe shi]> to tlie ig l i t was 7*. A short time later, at pijint D, the angle ofeleration was \(C

!25tt

To the nearest foot, determine and state how far the ship traveltxl IVotn point A to pant

Topic 3; Circles

Definitions and Theorems:

An inscribed angle is one whose vertex lies on the circle and whose rays both intersect the circle.

An inscribed angle is equal is equal to one half the intercepted arc

A central angle is one whose vertex lies on the center of the circle and whose rays both intersect the

circle.

A central angle is equal to the intercepted arc.

The angles formed by intersecting chords is given by one half the sum of the measures of the

intercepted arc.

A tangent line touches a circle in exactly one spot

A secant line is a line that intersects a circle twice

The angle formed by tangents and secants Is given by one half the difference of the measures of the far

arc and near arc.

The angle formed by a tangent and a chord is equal to half of the intercepted arc

S E C A M S E C . M E M L E N C I H TiiEOkEM PA-PC

!f two secants are drawn from a common exterior point, then the product of die icagtli of one secant with the length ol" its external segment is equal to the product of the length of the other secant with the length of its e-xternal segment.

8EC .AS rrf ANGEM S E G . X I E M L E N G l i ! TUEOItEM

If a secant and tangent are drawn from a common extmor point, then the product of the length of the secant with liie length of its external segment is equal £a the square of the length of the tangent.

I . N T E R S E C T E N C CtiORDS ANI> S E C M E N T S

i f two chords intersect within a circle, then the pnxluc! of the partitioned segii.ient.s ol'one chord equals the prtxluct ofthe panitioned segnienLs of the other chord.

Examples:

|. In circle M below, diameter AC,chortls AB ;md B C , ami radius MB are drawn.

Wbicb statement is not tme?

(1) A A B C is a rigbt triangle. (3) m B C = mCBMC

(2) A A B M Is isosceles. (4) niAB - - - m A A C S

Z . In tbe diagram below of circle O, cbord DF bisects chord B C at £ .

D

I f B C ~ 12 and F E is 5 more than DE, then E E is

(1) 13 (3) 6

(2) 9 (4) 4

3 ' Quacirifateral ABCD is itiscrihed In circle O, m shown below.

I f i n A \ mC mCB = 757 m A C = {y + 30)7 and m A D = (x ~ 10}°. wii icb statemeii! is true?

{!) X - 85 and r; - 50 (3) i - 1 iO and fj = 75

(2) X = 90 and = 45 (4) x = 113 and t/ = 70

•V« b i tbe diagram below of cirole O, chord CD Is parallel to diameter

JOB :md m C 5 =

A

What is inAC':^

{i) 'IS {3) 65

{2} 50 (4) 115

6- i l l the diagram sha\VB htdow, PA h taiigont to d r d e T at A, and

set-ant PBC is drawm where point B is on circle T.

If E B = 3 and BC - 15, what is the length of PA?

(1) 3v§ (3) 3

(2) 3 ^ (4) 9

I n the (J iagrambdow of d r d e O, OB and OC arc radii , and chords AB, BC, and A C are drawn.

W h i d i statement mast always I K S tnie?

(1) ABAC ^ Z_BOC

(2) mCBAC = JmAHOC 2

(3) A B A C and A B O C are i.sosedeN. (4) The area of A B A C is twice the area of A B O C .

-]. I l l the diagram below. DC, AC, DOll CB, and AB are chords of circle O. F D E is taegmit at point D , and radius A O is drawn. Sam decades to appiv tins theorem to die diagram: Ai\e inscribcal in a semi-circle is a right angle."

C

I f in ABCD = 30°, dctennirie and .sfate mZAOB.

^- I'll circle O slicnvn liclovv., diameter A C is perpendicular to C D at i ioi i i t C\d cli<irds AB, BC, AE, and CE are drawao

Wl i i c l i statement, is not aKva)-s true?

(1} ZACB ^ ABCD (3) EBAC ^ A D C B

(2) A A B C ^ A A C D (4) A C B A = A A E C

Topic 4: Rigid Motion

Definitions:

T R A N S F O R M A i T O N S I N T H E P L A N E

A transfyrmjiti&n, F, is a function (or niie) that for every point, P, in the plane as us input gives or assigns another, single point in the plane, F(P), m its output,

R O T A T I O N S

We can rotate a point .-i in the plane by an angle of ft around another point fS by constructing segment .4' B such that yVB = .IB and mZJ' BA ^ 6. We say. Ay 9

«-d-f) = - T X \^

P R O P E R T I E S O F R O T A T I O N S

( A N D A L E OTHER RtOiD M O T I O N S )

1. Rotations transform lines into lines, segments into segments, and rays into rays. 2. Rotations preserve the distance between points and hence the length of line segments. 3. Rotations preserve angles between lines, rays, and segments.

Geometric Fact; A line which is rotated ISO about a point not on the line will always result in another line thai IS paralJei tp the prigiHah A line which is rotated tKOA-ibout a point that is on the line will simply produce the same line,

R E F I T X T I O N N

VVe can reflect a point A in the line m to produce A \. { A ) ~ A \y using the following:

1. If .1 does not he on w (hen UK- :UC point A' on the other side of/n such that segment A.-T siuisties: A

a. AA'Am and b. _TJ" imersecSs line m ants midpoint

2. If A lies on m then its reflection is itself, i.e. ( A ) ~ A

T R . 4 N S I A T I O N P R O P E R T I E S

1. Map Sines to parallel lines (only true of translaiions). 2. Preserv'e angles (true of all rigid motions). 3, Preserve knglh-'dislance (inie of all rigid motions).

C O N O R ! E-NCK Two figures in the pliuie arc tongrutnt if a sequence of rigid motions can be found that make the two figures coincide (or iie exactly on (op of each other). We call this sequence of rigid motions a congruence.

pROPF.RTirs O F Riran M O T I O N S

1. All ligid motions map lines to lines, segments to segments, and rays to rays. 2. All rigid motions prcsciwe distance and angle measurements. 3. Rotation of a line IHO about a point not on (he line produces a paralleliine. 4. Rotation ofa line 1 SO' about a point that does iie on the line produces the same line. 5. Translation of a line not along the line produces a parallel line. 5. When a point is reflected across a line, the segment connecting the point and its image is perpendicularly

bisected by the line.

Examples:

J. in the tliagratn below, u seqiienco of rigid motions maps ABCD onto JKLM.

y

If mAA - H-27 mCH AM is m7 and wAL = mig the measure of

2 . The graph below shows two eongnicnt triangles, ABC and A'B'C.

y

, II \

f 1

! • E j j

....

T r FA \ c ; ^ \ 1 c ; ^

A V

B .J

B

c >

1 i

which rigid motion would map A A B C onto A A ' B ' C ?

(1) a rotation of 90 degrees counterclockwise alxjut the origin

(2) a translation of three units to the left and three units up

(3) a rotation of ISO degrees about the origin

(4) a refleetion over the line i{ = x

) The vertices of A B p B liave coordinativs B(2.3). Q(3,H). and B(7.3). Under which transfnrniation of A B p B art distanee and angle ineasnre nreser\ed? r (1) (x,7)-*(2.v,37)

(2) (x .7) -*U- 4- 2,37)

(3) (xjj) - * (2v, (y + 3)

(4) (.V.7) (x 4- 2.y + 3)

h. A A B C congment to AHST? Use the properties of rigid motions to expiaiii vour reasoning.

6. Triangle ABC and triangle .ADE are graplied on tin- .set of axes below.

y

j j 1

i

D. D.

1

I i

« 1 E t—-|-^ 1 E

HBi HBi sLe

y j y r I \

D!xscribe a transformation that maps triangle ABC onto triangle ADE.

Kxplain why this transfonnution makes triangle A D E similar to triangle ABC.

t). The image of ADEF is AD'E'F'. Under which transformation will the triangles not ho emignumt?

(1) a refleetion through the origin

(2) a refleetion over the line // = x

(3) a dilation with a scale laetor ol 1 centered at (2,3)

(4) a dilation with a st ale laetor of g centered at the origin

Z. 3\h tran.sfonnation wonlil not alwavs produce an image that would be eongnicnt to tiie original figure?

(1) translation (3) rotation

{2) dilation (4) renection

As sh tl ie quadrilateral is a rectangle.

M'hicii translonnation w^oiild not map the rectangle ontti Itself?

(1) :a reflfrotion over tiie y-axis

(2) a reflection over tl ie l ine a = 4

(3) a rotation of 180°* about the origin

(4) a rotation of .1.80 about the p r i n t (4,0)

A regular decagon ss rotated n degrees about its centei; carrvi tlic decagon onto itself. The value o f n could be

(.1) 10° (3) 225°

m 150° (4) 252°

Qiiadriiiitcral MATH and its image MW'T'TH are graplied on the set. of axes below.

y

A hZ-Iljl.

r

ifC-;--

X

Desciilie a .stujuenee iA transformations that maps tpiadrilateral MATH onto qoadrilateral MWrii".

In tlte diagram ljel(n\ ot A A B C and AXl 'Z , a 5ef'|neriee of rigid motions maps ZA onto ZX,

Detennisu* anil state wiiether BC = f Z , Explain why.

If hi the tliagni-m below, A A B C ^ ADEf\

y

W'Tiich isequeuce of transformatiori-s maps A A B C onto ADEF?

{1) a reflection over the x-mds followed by a transiation

(2) a reflection over the 7-iixis foiiowtKl by a t.ranslation

(3) a rotaitoit oi IH(C aliaut the origin tollovvi:^! I>y a traiislatirin

(4) a ct)iJLnterdock%vise rotation o f 9 0 ° about the origin foUowe< a trairslarion

In the two distinct acute triangles ABC and O l i i q AB = Triangles ABC mid DBF are congnjent when there Is a seqnenc' rigid motions that maps

( I ) AA onto Z D , and AC onto AF

{2} AC onto "DF, and BC onto EF

(3) AC onto AF, and BC onto EF

(4) pohi l A onto f^^hit O, and A B onto DE

- Triangle ABC and triangle DEF arc drawn below.

It AB s DE, AC = DF. and A\ AD, write a sequence of transfonnations tliat maps triangle ABC onto triangle DEF.

Triangh- ABC has vertices at A(—5,2), B( —4,7), and C(~2,7), and triangle D E F has vertices at ' D(3,2), £(2,7), and F{0,7). Gr^di and label A A B C and A D E F on thi- svi of axes !>elow.

Determine and slate the single transformation wlicrc ADEF is tiie image of A A B C ,

Use \xnir transformation to explain why AABC ^ ADEF.

Identify which sequence o f transfonnatioiis could map pentagon ABCDE onto pentagon A^BTPCEf m shimm Mow.

(1) dilation followed by a rotation

(2) tniuslation followed by a rotation

(3) line reflection fol lowed by a translation

(4) tin© reflection followed by a line reflection

/Z, Tlie graph below shows A A B C and its image, AA"B''C'F

A"

l>eseribc a.sef|uonc€ ol rigid motions which wiiuld map A.ABC ouio A A " B " C " .