Geometry - Unit 4 Torgets & Info Nome:This Unit's theme - Proving Triongles CongruentApproximotely October ?8 - November 19Use this sheet os a guide throughout the chopter to see if you are gettingin reaching eoch torget listed.

By the end ofknow how to...

Identify and use correct vocabulory: SSS,SAS, ASA, AA5, l{1, CPCTC (what does itstand for), postulote, corresponding ongles,equiangulor, equilaferol, ouxiliary line,olternote interior ongles, congruentf riongles

Unit 4, you should

Determine if triongles ore congruent usingSSS, SAS, ASA, AA5, ond HL

Colculote the measures of sides and onglesof o triongle using the isosceles triongletheorem and its converse, 180" in ofriongle, and equiloterol triongles

Target foundin...

Chapter

Prove triongles are congruent by providingstofements ond reosons to complete opartially completed two column proof

D¡d IreochtheT oroel?

the right informotion

Complete a blonk two column proof oboutcongruenl triangles using given informotionond o diogrom.

Chapter 4Sections ?-3, 6poges 2?6-243

258-264

DIAGRAMS &EXAMPLE S !

Chopler 4Section 5,poges ?50-256

Chapler 4

Chapler 4Section 7Pages ?65-271

Classifying Triangles:

I. Side Lengths

Lesson 1: Congruent Figures and Triangles

Equilateral

Angle MeasurementsII.

Be as specific as possible:

Classify:

Acute

Isosceles

Right and Isosceles Triangles:

Equiangular

\Right

Theorem: The sum of the interior angles of a triangle is

Scalene

{---}<l.-}Given: AABC, AB ll CDProve: mLl + mL2 + mL3 = 180

Statements

Obtuse

Corollary: The acute angles of a right triangle are complementary.

Theorem: The measure of an exterior angle of a triangle is equal to the sum of the measures of thetvvo nonadjacent interior angles.

Congruence in Triangles:

Writing Corresponding Parts:

Proving Triangles are Congruent:

Yonr l¡arr for prlr Given: Tli = ¿t¿ ,ltN = @of congrueat ptl¡,Wh¡'t sl¡s do yo¡l¡eed to prove teuimgler cofignænt?You need ¿ ûilÚ pôF of(mgn¡ent lides ¿nd athíd par of congnuenlañgltr

¡: lll = ¡ (), ." ,tll.,\ := ¡ (J1.ñ

Prove: it,,lf,\' =,:i ¿(/,\'

p.222 L0 - 20,30 - 34,36, 38 - 40,45,54,57,58,

Stetemcnt¡

Ãi = t(,, òw-= Nt-l = ¿,V

; .lll - ,: (1. 1 ,lfl ,\' - , (Jl ,\'., IrVr- .r: ,' (_),\'r.

,', l.i\1,\' -= .'.1.1-),V

1)

r)3)

r)5)

1)

r)3)

{)5)

Lesson 1 Practice: Congruent Figures and Triangles

Match the triangle description with the most specific name.

l. Side lengths: 2 cm, 3 cm, 4 cm A. Equilateral

2. Side lengths: 3 cm,2 cm, 3 cm B. Scalene

3. Angle measuresr 60o,60o, 60o

4. Angle measures: 30o, 60o, 90o

5. Side lengths: 4 cm,4 cm,4 cm

6, Angle measures: 20o, 145o, l5o

Classify the triangle by its angles and by its sides.

7/À

ÁÀ

C. Obtuse

D, Equiangular

E, Isoscelest F. Right

Complete the following statements using ølways, sometlmes, oÍ never,

N

13. An isosceles triangle is

14, An obtuse triangle is

11,

15. An interior angle of a triangle and one of its adjacent exterior angles are

9.

16, The acute angles of a right triangle are

17. A triangle

an equilateral triangle.

12.

an isosceles triangle.

has a right angle and an obtuse angle.

complementary.

supplementary,

Find the measure of the numberçd angles.

mZL =

The variable expressions represent the angle measures of a triangle. Find the measure of e¡ch angle.Then classify the triangle by its angles.

20. mLA = xo 21, rnLR = xo 22. m¿W = (r - tS)"rnZB = 2x" mLS = 7x". rflLy = (Zx - 165)"mzC = (2x * LS)' rnzT = xo mLZ = g0o

Find the measure of the exterior angle shown.

)1

mLI = mLZ =

In the diagrnm, AABC = ATUV. Complete the stntement.

25. ¿A =26. VT =

mL3 =

27, AVTU = _28. m¿A: ntL_=

8cm

J-dentity any fÌgures that can be proved congruent. Write a congruence statement.

32,*;

-ü;

35, Find the midpoint of AB for A(-2, 8) and B(4, -3).

33.

36, If M(-3, 6) is the midpoint of XY, and the coordinates of X are (7,1), frnd the coordinates of point Y.

37. For A(5, -2) and B(8, 4), flrnd AB.

,t

TF38.TF39.TF40.T F 4I.

TF42.TF43.TF44.

If a conditional statement is true, its converse if false.

If a conditional statement is true, its contrapositive is true,

If the inverse of a conditional statement is false, then the converse is also false.

If a statement is true, its negation is false,

If two angles are congruent, their supplements are congruent.

If two angles form a linea¡ pair, they are supplementary.

All right angles are congruent.

45. mZl =

m/3 =

mZS:

SJ¡te which lines' if any, can be proven parallel to each other with the given information.

m/2=

ml4 =

ml6 =

47, ll = 1948. 25 = Zl049. 17 = lll50, l8 = Zl2

4x + 13¡"

46, x=

v:

51. 13 = Zl452, 23 = ll253, m Z13+ml14:180o

54. sJ-wandul.w

55) Oiven: ll = /2,13 = 14

Prove: n ll p

Lesson 2: Proving Triangles are Congruent Using SSS, SAS, ASA, AAS

All six parts of a triangle are exactly the same as the corresponding six parts of another triangle

nx: AX\ÂrY ÉAvzx

1) SSS (Side-Side-Side)

If three sides of one triangle are congruent to the corresponding three sides of another triangle, then thetwo triangles are congruent.

2) SAS (Side-Angle-Side)

If two sides and the included angle of one triangle are congruent to the two sides and the included angleof another triangle, then the two triangles are congruent.

3) ASA (Angle-Side-Angle)

If two angles and the included side of one triangle are congruent to two angles and the included side ofanother triangle, then the two triangles are congruent.

4) AAS (Angle-Angle-Side)

If two angles and a non-included side of a triangle are congruentto two angìes and the non-included sideof another triangle, then the two triangles are congruent.

Name the included angle between the pair of sides given.A

1) AB and CB

4 Ñand sD

3) AE and CE C

Determine if there is enough information to prove the triangles congruent. If yes, then state the postulateor theorem that would prove them congruent.

Given: AB = CD,ABIICD

Prove: aABC =yCDA

s)

Statements

A

/

D

Is it possible to prove the triangles congruent?prove it.7)X

Given: AD ll EC, BD = BC

Prove: AABD ÈABBC

If yes, state what

B)

Statements

postulate or theorem you would use to

10)

Lesson Z Practice: Proving Triangles are Congruent Using SSS, SAS' ASA.

Name the included angle between the pair of sides given. J

l. lr an¿ Kt 2, JL and iR'

3, PT andffi. 4, -KL and JL

5. LP anLK

Decide whether enough information is given to prove th¡t the triangles are congruent. If there is enoughinformation, state the cong¡uence postulate you would use'

7. AUVT,AWVT 8. ALMN,ATNM 9.

6, KP and PL

AAS

L.

I I. ARST, AW\TUT

t.ffi=ffi2. FG =FE3.m =m4, AEFG = AGHE

Statements

aYzw,^Yxw

12, AGIH, AHLK

l.

2.

3,

4.

14,

15.

l.

2,

3.

4.

otvEN > frF- d-E =fr,, ¡fr=fr :PRovE >

^NPQ = ARSr i

l.

2.

3.

4.

16.

otvEu>F=To,-t4[øPROVE>AABC=A,CDA

otvEf{ > PZ bis.ecrs L'S/T,SP =TP

PRoVE>.A,SPQzATPQ

'P e'

17. Statcmcnts

18.Statements

PROVE > AIQT= ARS1

P

t9. Statements

olvEil )E = E?,Mis tbe midpoint ofTB,

PBOVE>AACM=ABCMc

GtvEil>ñ=Æ,,F0=þ,ffi=-rc

pROVE > ^ABC=

A,BltE

For #20-22, use the points A(-2, 5) and B(6, -l)20. find AB

2t. find the coordinates of the midpoint of eg

22. find the slope of the linelË

TFTFTFTFTFTFTFTF

23. The acute angles of a right triaagle are complementary.

24. A theorem is a statement that is accepted without proof.

25. The converse of "If p, then q', is ,.If q, then p,"

26' If a conditional statement is true, Íts converse is false.

27. The inverse of "If p, then q" is ..If p, then not q."

28, If a statement's converse is true, then its inverse is also true.

29. The contrapositive of "If p, then q" is ,.If not q, then not p,"

30. If two lines do not intersect, they are parallel.

31. Use the distance formula and the SSS Congnrence Postulate to show that AABC = ADEF.

32. In aRST and aABC, R5 = Æ, s-T =ñ, and ffi' =m,. which angle is congruent to ¿T?

(A) zR (B) ¿A (c) Lc (D) cannot be determined

AABC = AXYZ

zA=AB=

A

^t-

Lesson 3: Using Congruent Triangles

CPCTC:

Corresponding Parts of Congruent Triangles are Congruent

ZB=BC=

v\,/

X

z-L =

AC=

Given: FL¡\m,W =ffiProve: FL = WH

Statements Reasons

Auxiliary Line: a line added to a picture to help with a proof

Given: AFGH is an isosceles triangle with vertex ZG

Prove: ZF = ZH

Statements Reasons

ISOSCELES TRIANGLE THEOREM:

If two sides of a triangle are congruent, then the angles opposite from those two sides are also congruent,

650 Xo

x= y= x=

L_

y=

THEOREM

Given: LF = ¿H

Prove: fr = nG

CONVERSE OF THE ISOSCELES TRIANGLE THEOREM

If two angles of a triangle are congruent, then the sides opposite those two angles are also congruent.

63o 63o

x=

x=

y=

z-

x=

Given: ¿L = L2, L3 = L4

Prove: -FC =W

ReasonsStatements

Lesson 3 Practice: Using Congruent Triangles

4x+ 72

x=

x=

y=

L_

x=

y=

x=

y=

x=

Given: AB = AC-->AD bisects LBAC

Prove: BD = CD

Statements

AB=AC

LB=LC

+AD bisects LBAC

LL* L2

AABD = AACD

BD=CD

2.

3.

4.

5.

6.

8. Given: Lt = L4

Prove: AB = AC

Statements Reasons

9. Given: RV llST

QS =QT

Prove: LL= L3

Statements

10. Given: Ll = L2, L3 = L4Prove:A BCE = Apcg

Statements

D

CÁl'\T

Reasons

43

A

Lesson 4: Hypotenuse-Leg Theorem

Can you prove that the triangles below are congruent?

AX

Hypotenuse Leg Theorem (HL)

If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and a leg of asecond right triangle, then the Wvo triangles are congruent.

Are the following triangles congruent? If they are, state the reason you would use to prove themcongruent.

T.

4.

2.

NN5,

3,

6.

PLEASE NOTE: To use HL in a proofi, you must have

1) a right triangle2) the hypotenuses congruent3) a pair ofcorresponding legs congruenT

Given: ÞFr q-ñ.

PSIQS

-Qn = QTAPRQ = APSQProve:

Given:

Prove:

ÞFr qT

ÞSr qT+PQ bisects ¿SPR

^PRQ = APSQ

S

Àa

Lesson 4 Practice: Hypotenuse-Leg TheoremDetermine if each pair of triangles is congruent. lf they are, tell which postulate or theorem you could use to provethem congruent. Mark the picture with congruence markings if necessary,

2.

7.

5.

3.

L0

8.

6.

-l

o

13. x=

L1..

9.

L4. x-y=

L2.

15. x=

y=

16. Given: LW is a right angle

ZY is a right angle

Prove: WZ=YZ

WX=YX

Statements

ZW is a right angleLY is a right angle

LXWZ and AXYZ are right triangles2.

3.

4.

5.

6.

WX = YX3.

XZ =XZ

LXWZ = LXYZ

WZ=YZ

Given: LF = KF

LA=KAProve: Lf = Kf

Statements

L7.

Reasons

4.

5.

6.

l,e = Xe

LA-KA

FA=FA

AFAL = AFAK

LL=LZ

FJ=FJ

AFJL = AFf K

Lf=KJ

2.

3,

4.

5.

6.

7.

(2x - t2) ft

18. x--

2L, Given: gF f EG

t9. x=

y=

z=

EH=GF

Prove: LH= LF

HG J- EG

Statements

20. x=

y=

Reasons

Unit 4 Test Review

Tell if each pair of triangles can be proved congruent. If the answer is yes, state a reason you wouìd useto prove them congruent. Mark each pictures with the appropriate congruence signs.

4.

2.

NNA5,

3.

10.

B.

6.

LL.

9.

L2.

13. x= L4.

L6.

X+! * Z=

x=

15.

37 cm

L7.

L9.

x=

x=

x=

y=

18.

22.

20.

x=

x=

x=

y=

21.

23.

x=

x=

y=

TFTFTF

24. A corollary is a statement that is very easy to prove from a theorem.

25. A theorem is a statement that we accept without proof.

26. If two sides of one triangle are congruent to two sides of another triangle, then the thirdsides are congruent also.

27. An equilateral triangle is also equiangular.

28. We can draw auxiliary lines in a diagram to help us with a proof.

TFTF

29. Name 5 ways to prove triangles congruent.

t. z.

4. 5.

30.

3L.

CPCTC stands for

The measure of the vertex angle of an isosceles triangle is 118'. What is the measure of a baseangle?

32. InAQRS, LQ- LS,QR= 3t+ 4, RS = 5t-8, and QS = 4t- 12. Find tand theperimeterof thetriangle.

Y33. Given: XY =ZY

Prove; L[= L4

3.

Statements Reasons

34. Given: LL = L3

RX=RY

Prove: ARXS = ARYS

Statements

35. Given: AM = CM

AB I- BD

Prove: LA= LC

Statements

CD]BDM is the midpoint of gD