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A: Summary of Main ldeas TRIANGLES: What are the types of triangles and their properties? How do we prove congruent triangles? How do we apply the properties of congruent triangles to determine information about parts of a triangle? How do we use the special segments of triangles to determine information about parts of a triangle? Terms you should KNOW! Unit 1 & 2 Terms triangle vertex plus: adjacent sides opposite side interior angles corollary scalene triangle acute triangle obtuse triangle isosceles triangle base base angles exterior angles vertex angle perpendicular bisector angle bisector legs hypotenuse right triangle equilateral triangle equiangular triangle midsegment corresponding angles corresponding sides equidistant Be sure you can classify triangles by sides AND ang les. Ct fbtr,i 3r s r a'cil(t\Q- tulc *Carlrrre {i,,,ilafetaL fiduI0 r^iq $ciie're-o *'* t'$"''wj^ta{ f* bJu :*- ,t*'u, Other lmportant Triangles Information Right Triangles and lsosceles Triangle?_ B* HYPotenuse: f'5C Vertex: r-.g' Base A15 r-eg: y'\L 6 Angle: A Base Angle: c AZ riLeg: f+ ti ;G Leg: eur", L f)L: c Gorollary to Triangle Sum Theorem m/.A+ mlB + mlC -- mZ.A + -$q' mlB : '1a" Third Angle Theorem Exterior Ang le Theorern _wLA +rnLh B A\ tfl/A =rfir-D "X \o ,Xi--.-\o Congruent Triangles and Figures Congruent Triangles SIDES ANGLES AB=SE LA =ZD ZB=L€ BC = h-= €:F A D 't/'l --, *\\ t::-.'\ F,, .''' Ll=tF llL = DF *\--. =#.- --l,''''''''''''''''''''''''''i Naming Gongruent Figures Properties of Triangle Congruence Given that ABCDEF = HJKLMI'I Reflexive ProDertv: LABC = List all pairs ofcongruent sides: m, ? r--u_ _ E:, )ii, Ifr A Al;C' Symmetric Property: i unc = 't-f [u f='t"". R ki..j-r< ^AF"=H\J. Llst all pairs ol'congruent angles: LAitn t$*= t-L L fi-=L-T / E"' L L(-'z t u Ltr'L t-xt ^DEF,th"' A .Dtf : AAO0 Transitive Property: If MBC = A,DEF 14 ANd LDEF = AJKL,thE" A AdC g AsK L Proving Triangles Gongruent We need at least 3 pairs of congruent corresponding parts to prove triangles congruent. The six possible combinations are below with the four that don't work x-ed out. Side-Side-Side Side-Angle-Side Three pairs of congruent sides Two pairs of congruent sides and one pair of congruent angles (angle between sides) hA Angle-Side-Angle Angle-Angle-Side Two pairs of congruent angles and one pair of congruent sides (side between angles) Two pairs of congruent angles and one pair of congruent sides (side not between angles) f-\ ,,f-\ n--- ,s:--- Base Angles Theorem Gonverse to Base Angles Theorem B B lfAB=BC IflA=lC 'rJ /n-l/' then ./- 11 - t-t- then I4b ,A r rv n - = f)U Gorollary to Converse to Base Angles Theorem Corollary to Base Angles Theorem If a triangle If a triansle is eouians.ular. then it is then Hypotenuse-Leg (HL) Theorem Perpendicular Bisector (J- bis.) Theorem t\ t\ t\t\ it i'" il e4u'il ii U aL Converse to the Perpendicular Bisector (l- bis.) Theorem ? \\ P L-\t-\ K-L B-c If O Remember, to use HL you MUST have f'rcih{ / 5 C Angle Bisector (l ! bis.) Theorem is I bisector of AB, then AC=BC Converse to the Angle Bisector (Z bis.)Theorem If AD = BD, then D is on the bisector of AB I Midsegment Theorem R /\ D+€ AC Tf AD iS Z biS, Of IBAC then D-B : DC , If DB: DC,then bis. ADrs I Triangle Inequality Theorem Ordering Sides Ordering Angles DEllACand DE=+AC of IBAC U M i.{ z-l 9 Lh F T l+l . !3 . *Id Hs >FU UN ul- *F N ', utu W * {) Ir/' o, Proofs l. E ,qC; nA nC = = Prove: a,4RE =^ARC Given: Statement Reasons , Tv=- ghu LR ZKC 2 [A A-r. , : AAKE A A(C- " t /otvU',;. "T?"l lerrv<, 3. 5:: 2. Given; B is the midpoint of AE and Prove: yABC =aEBD Statement rbr:6r/pt {Ya G z 3 rAsc :4 OsE 4 AAhL !AEbD t.6iw^' i. [,{ 6n'dpo'nt 3 Venhtco 'L5 4. 5A5 3. Given: Pf m; ITPR= IPRA = Prove: {RP =yAPR FT lAA , mYm Arnp LTPRY- TPRA r A APR 6tv^"-* q* e_+le { l\ 545 4. Given: fd = S,q; ,qf is an angle bisector 4LYL3 Statement 1fn=S,A? L bi:rclar 2 R"=ffi s Zl5AT o /t1r ArAt 5,i Le* L 5"ASAT t (t}'e'n: i, {e lle*w 4 ef L br:ccfor {, 5A5 CPLTC 5. Given: ,qnlt nC; An nC = Prove: aABD =aCDB D L-' Statement Reasons 1. z. J. 4. Gtw* t At+ l'4+ Ls f& {letii"<-- 5.A3 C: Problems t. AB= BD . .qn = Uf :-: i{\\hLrlr4'" (\LI if fn(6ltcrn'.> Dr ) a1 L hr *etnr Atr J- hrSctrrr:; &irm : Name all segments that are: a. altitudes, b. medians, c. angle bisectors d. perpendicular bisectors \ 2. J. Two sides of triangle measure 5 ft and 13 ft. Find all possible measures of the third side. a 1t ^L tt ALNG is isosceles triangle with vertex L w1.ANS =35o m,/1 - J -t ,73 = AO 4 SP is angle bisector of IQSU Pf is perpendicular bisectorof trS PR ir purpundicular to QS IQ, ir XS= ll mZpSR = -an rtlrr- UT= l"l PS= lu 4..l J I t' y', PX= U w 14 o ,zz =45 ,2a= 50 Multiple Choice Practice D: Ee Multiple Choice Grven LX LZ = LO, find, the value of =.r l/ and $. r. Muttiple Choice Whrch posrulate or rem can be used r rheo_ "qiffiF#;;iJffnT"ni."*:ru @ SSS @ SAS (pe) (O38 (A) @zo @) ssn G) AAA c)95 L 6, lfuttipte Choice W_hich starement correctly congruence of tl_," ,.r"ngl", ,n $:,r:rtb"rlhe . ,b,XercrSe 5? @ d. Multiple Choice Use the diagram rn Exercrsej to find mLZ @ lg' @38. @ 70. €"e @ 95" LKML= ApeM @* AKIM = [.peM , (e:@AD CD AKLA4: LpMe AKUL = Lpeta @ 7^ Muttipte 3. MuLtiple Choice If I,ABC = LXyZ,whrch of the followrng statements below rs not true,/ @ LB=Ly (-\ f -e*-.'""-'..'.-"..-'\ r-D^ W@ - LRAC @ TE=Xy ,. ae=YZ = Lyxz ;:ltfr'j:: i-!*"*' proving the tnangles ;;, =pet ; ;;:; .'Ji: Verhcal Angles ,Itreorem 9 P-*!=". @ C) rf Multiple Choice LTse the dragram below Which congruence rs correct to prove AXYZ = A]KL? Choice congruenr," 5 ",.-^*^'P You grve ro prove KL iprnp oI Cong^.n.. ASA Definitron of mrdpoinr 8, Muttiple Choice Whrch posrulate or rheorem can be used ro prove rhat LnfC ii?cz = X D\ A\ z--t\ --a @ Ly = LK by SAS Congruence postulare -p*g:g*u!{s*gqlerq-"s.rio',:lgg((e-) ZY = LK br SA: Congruence t;,r$ C-. B orC @ (g@ SsS @ SAS tFl @ none of the above ASA Mut4ple Choice Whar is the thrrd congru_ ence needed to prove rhat AABD = ACBD by AAS? Q. 83. &ruhpte Chobe @ x: <d;;=:qBD l;\ r=::-- lO. and ) 120, @ ":60, y:60 O x: 30, LtL @ LDBA = LCDB @ BorC r y:6O (A) ,qB=fC /1L/: Solve for y=120 A Multiple Choice What rs the thrrd congmence needed to prove that LMNe = Ap1/O by ASA? @ _e=tp aA) A/^r/^\ -i 11 *'tt.vv - 14. futultiple Choice Solve for x and v @ x: J0,), : 55 tE, -l = 55 rr : ttr'r M / -P/VO (D /M= Lo /.K'--;- -:;\ watr) .g lE" r.]v\ _,ya1 ==--j-.-:z*.._-**j-y vn @ x=70,y:145 @ x:55, y:145 O t\v Multiple Choice Grven rhat LX LD, and = DE = Xl{r, what rs the third congruence need_ ed to prove rhat AXI4IF ADEb bv ASA? = Gl Ly= LC O Lw= 29 L] b. Multiple choice @ LY: LE (A) x: 72.5,y @ x: j5.,: <D -w=LE ) "-. none of the above \-\z_/ -%--------- (D) /-E-) \-*Y { lQ" Mulnple Choice Grven rhat LG = LE and LI : LD, what rs the third congruence need_ ed ro prove that AG111 = AEF bv AAS? @ /H: r-n\ \12-l Hl: @ tD FD ) @ Hr =ED ED =ej What are the values of x and y? J-) j ,:35,|: tln 55 x: 110,): J5 ,6, Multiple Choice In the dragram, ffi rs the perpendicular brseclor ol ,qB Strat is the value o1 r? none of the above z 16 (ffi> 10 1,. Multiple Choiie In rhe dragram. e KM and7P =* PE Frnd LM. --*\ ,.T; 8 ', {\-----7 @10 G) 72 @13 (E-) 1) 20, Muttiple choice Lrst the longest to shortest in order from tAl 7r Ei r.-. @ ) Lr , cLI EG,GF,EF Fr--..\,!4, Lf , LG, GF ffi, w \E I:U.tk,GF /f M- - t8. Multiple Choice Use the diagram below to frnd FG. srdes - 4u"y"Wr: Choice In the diagram, W are rnrdsegmenh of LXYZ Find t{T. andW-D @10 @15 @,4 12x+8 --_\ @3 .f @44 -_\ M G) 6 6l j L2- Multiple Choice Vlhrch posfulare or theorem can be used to prove that XenC = ABADT @ sss 14. Mutnpte Choice Use rhe dragram below Which addrttonal congrllence _rs correct to prove A,4BC = LDEF? FE @ @ LB = LE by SAS Congruence postulate FE by SSS Congruence postulate -BC = O tA = -D by SAS Congruence postujate @ AorB @ SAS --6, --:;: --Y--:j.W (D AAS @ none of the above A 73. nfuttiple Choice The tnangle below ciassified @ u.u,. as l rro*il @ acute scalene /-6-'T_ -:-*;-_ ru \Llr obtuse SCa.lene @ nght scalene can be Q4" rw"uiple choice A tnangle has rwo srdes rhal have lengths of g cm ard 14 cm. Which length below could nol.represent the length of tbe third srde I @ 7cm --*\ 22cm @) LLeM I 45. Muttipte @ 13cm O ) 1 choice (E, I) 18cm Cm What is the value of x? @3 2x+8 &&. wutaple Choice Whieh. tyee of segment need nob cotrta.in a vertex of the triangle? altibude b" angte bi,sector c. median