Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Dessin d'enfant wikipedia , lookup
Golden ratio wikipedia , lookup
Noether's theorem wikipedia , lookup
Four color theorem wikipedia , lookup
Multilateration wikipedia , lookup
Apollonian network wikipedia , lookup
Rational trigonometry wikipedia , lookup
Reuleaux triangle wikipedia , lookup
Euler angles wikipedia , lookup
Trigonometric functions wikipedia , lookup
History of trigonometry wikipedia , lookup
Euclidean geometry wikipedia , lookup
~lD(AY fv'ldNDfAY 'ley Name t Chapter 4 Study Guide -res, I\~S'OfAY Section 4.1 - Classify Triangles and Find Measures of Their Angles A triangle is a polygon with three sides . .A sca \~X)fJ triangle has no congruent sides. An isosceles triangle has at least An eCA\) \ \C\leroJ Y \ tjY\r congruent sides. lAlvrt M~\QCi2 triangle has one right angle. An obtuse triangle has one An equiangular D triangle has three congruent sides. An acut~ triangle has three A J\i\) ~W.5{;. triangle has three QY\~\t. Q)n~('J€.n1' OS)~\QS. When the sides of a polygon are extended, other angles are fonned. Yhe original angles are the angles that form linear pairs with the interior angles are the Triangle Sum Theorem: Exterior Angle Theorem: tX\f.X\~r In:k~n~u~;( angles. The angles. The sum of the measures of the interior angles of a triangle is , 0 The measure of an exterior angle ofa triangle is equal to the sum of the measures of~ nun Lld,\d(Qnt lO\tilL'f cU ~~h6 __ Corollary to the Triangle Sum Tlieorem: The acute angles of a right triangle are 1.,[\ 2/\ 3~ C.dY\~ Classify the triangle by its sides and by its angles. 60 4. FindmLABD and mLBDC. .Find mLCAB and mLCBA . • A 17 n~\\ts;~\~ne, eqU\Ii'\OS eq~\o.\f.m\ 3x.~2X+30 X ~ 30 .., V ~ :;" Section 4.2 - Identify Congruent In two congruent S. (Y)LA€iO ~/3 6) ~~~ ~. lE.J 2x ~ 2(?O}~~ , L f7DG Figures figures, all the parts of one figure are congruent to the corresponding CAB \ TZ:?\ tt;x ::t{{c) ~ ML IY1 L Q {\ parts of the other figure. In congruent polygons, this means that the corresponding sides and the corresponding angles are congruent. Third Angles Theorem: congruent. If two angles of one triangle are congruent to two angles of another triangle, then the third angles are In the diagram, LABe "" .6.DEF. B 1. Find the value ofx. 2. Find the value of)'. D (5y- 3)0 ~- A~C E F fjjjJ ~~(t<l) , Section 4.3 - Use Transformations to Show Congruence A rigid motion is a transformation that preserves motion is also called an isometry. JtG' C~'(,', I JfM ill ,~ ,to" , ffBCtbfK\S I'QlI1Sv'rt \ ,and , and O,fe~ Y1ltItt]6YlS . A rigid are rigid motions. Describe the transformation(s) figure B. you can use to move figure A onto Tellwhether a rigid motion can move figure A onto figure B. If so, describe the transformationls) that can be used. If not, explain why the figures are not congruent. 3. ~ J nr\71C\'<!(\ IflO C Describe the rigid motion(s) that can be used to move figure A onto figure B. 5.~ Section 4.4 - Use the Side Lengths to Prove Triangles are Congruent Side-Side-Side (SSS) Congruence of one triangle are congruent to three Postulate: of a second triangle, then the two triangles are State the congruence statement for the triangles that can be proven congruent by SSS. 1. 2. 3. :\1 ( A~B D I \ \ C ,\ f G \J J es Section 4.5 - Use Sides and Angles to Prove Congruence In a ,igh! "iang!', th, ,id" adja"n! tn th, ,igh! ang!' "" call,d th, ~ The side opposite the right angle is called the ¥eVUUfthe Side-Angle-Side (SAS) Congruence S\dt) . are congruent to two COO~Wf:ot Hypote to the Postulate: use-Leg (HL) Congruence Jl¥~\enU and a right triangle. es If two •., and the included Theorem: If the Ie . and the included ~ an8LL --h.¥V~\cQ\DV of one triangle of a second triangle, then the two triangles are lea and a ofa second right triangle, then the ~are of a ri~ht triangle are congruent CM . Decide whether enough information is given to prove that the triangles are congruent using the SAS Congruence Postulate. 1. £iPQI; £iRQS p 2. fj.NKJ,MKAI J R Z wlY T (V- p& -= tvO {<.Q LfQT~LlC-GS ordM~ sPf::> QT ~ &S c.v' c:! S ~~ OJ'<-1"Ct~.sc.A.l'-t.. Section 4.6 - Use Two More Methods to Prove Congruences A JJ~ow~ Angle-Side-Angle proof uses arrows to show the flow of a logical argument. (ASA) Congruence Postulate: are congruent to two ~ CuY)f!lJt1lt Angle-ingle-Side If two and the includ~d . (AAS) Congruence triangle are congruent to two then the two triangles are ~~l2 Theorem: (0)~nt If two (si M angw> and the included .5\ fl of a second triangle, then the two triangles are . 1" and a non-in~luded and the corresponding non-included 51 ruG Adg.;---- Can the triangles be proven congruent with the information given in the diagram? If so. state the postulate Gr theorem you would use. 1. of one triangle of one of a second triangle, Section 4.7 - Use Congruent Triangles to Prove Corresponding Parts Congruent Tell which triangles you can show are congruent in order to prove the statement. What postulate or theorem would you use? 1. JK:::: LK 2. LRPQ:::: LTRS K p L J T M ~ PR.Q ~ L} tTS 4 J N\ 'F- ~ .6 Ltv\ r-- D'fS* 'v'fSfcS Section 4.8 - Use Theorems about Isosceles and Equilateral Triangles When an isosceles triangle has exactly two congruent sides, these two sides are the legs. The angle formed by the legs is the vertex angle. The third side is the base of the isosceles triangle. The two angles adjacent to the base are called base angles. Base Angles Theorem: If two sides of a triangle are congruent, then the Converse of Base Angles Theorem: to the Base Angles Theorem: Corollary to the Converse of Base Angles Theorem: 1. Find WY: opposite them are congruent. If two angles ofa triangle are congruent, t en the ~ Corollary Use the information given values. an ') If a triangle is 114U \lA:krv\.\ opposite them are congruent. ' then it is e~~;~n:~~~~.-4. 1 Ifa triangle is equiangular, then it is~. w in the diagram to find the x 0 2. Find mLWXY. 3. Find the values of x and y in the diagram at the right. I tJo - q 0 - b 0 -; 3 0~ R IS 2. ~ 12..'2. -t 11~ fA ~ ()... ZZS -=- 7. f7...2 :2. -t-(}••• fA-=- q 1-- --I u Section 4.9 - Create an Image Congruent to a Given Triangle JYC\US:ThimLtDW\ A is an operation that moves or changes a geometric figure in some way to produce a new figure. The new figure is called the image. -! ram k1tllN'\ A moves every point of a figure the same distance in the same direction. Coordinate notation: (x, y) A~ (x + a, y + b) uses a line of reflection to create a mirror image of the original figure. Coordinate notation: (x, y) m\I.ltlN\ A --+ --+ (x, -y) turns a figure about a fixed point, called the center of rotation. Coordinate notation: (x, y) --+ (-x, y) A congruence transformation changes the position of the figure without changing its Figure Sketch _.)_n_\U~ sh~ or ABeD has the vertices A(-5, 5), B(-1, 2), C(-3, OJ. and 0(-5.0). ABeD and its image after the given transformation. 2. Translation: (x, .v) --+ (.x + 6, Y - 2) Reflection in the x-axis 3. Name all the angles that have Vas a vertex. \, d-lV.r. L IV S LHV3 \T . 2. L \<-V~ j:..~ 'Vi-i-l.F ~ Al . 1\ K (Ex+em~y \, V 5. \10' "jX c •••• X~sn tlO :: V 6. Y V T C tAn.~\J2JWOY'eM ') 4. U 5ii .~ \ S LCVD LCV6 L. CVf L DVE L bVt Lf.VF 3. ~ J 0:\UJ Solve for x. YM /\ /15X+5\ EiOJ Q • I .4 no'/ 21H~ _ ) D B I Z(J -: \~X\-S- ~ 22-)(0i 12-0 .: 31 X III -= 57 X +4 G~?(--:.3-~ State if the two triangles are congruent. If yes, state how you know. 7. ~ 8. \ ~ \ b''j AS f\ (K\L CM--L- U\'l'i~ OJll;{'u..; \ &))1 ~&..o,~ f"jl~~'" --w Sl»1 V ~51\S t tJ,V\.~ -\ 1l'~\ State what additional information is needed in order to know that the triangles ar congruent for the reason given. 10. SAS 11. SSS 12. ASA f) --- Write a proof. 5. 4. GIVEN:AB==DB,BC-lAD PROVE: /').ABC == GIVEN: mLJKL = mLMLK= 90° JL==AfK /:).DBC PROVE:/:).JKL B == /:).}vfLK J (j) Be.. ::: - (\J I"}f p""' D 11 ~BC ~b(X?L IV'I L JR :: vYlL MLf-:: qOo (2) 0L ~ ~ A ..c:BCA M\AL.Bc.o ) ar-t- n~~ W'j \l; :.J M) ~,)\) OM- ) AB ~ Dl?> , ec....iAD ! 9. 3\vtt' ~n t\L CL~,~'V "IT-, ~ M(j) L\J\cL.~ lA-- brv\Lt-- ~M(\ rrdk)(,\it 1_ i:'\ c.~w-ti- ~~ HL CMj~tu... ~ck~~~~ r-th~w..{!h.~ (j) Cj~ 5~ (J.; 4. GIVEN: LPQS=:::. LRQS 5. GIVEN: L OltfN =:::. L Olv'Al LLlvl0 =:::. LJ/!lO LQSP=:::.LQSR PROVE: 8.POS ......, 8.ROS ....•.... =:::. PROVE: 8.A1.IN~ !:::.NUI M Q p R L s CD LPQS~ LgQ'S @ J 'D~'fI ~ &s~~ ~ lJ rG-S ~ LJ~.s (i) ~M-~~~) AS~ ~'f'J'lf'\£J-, ~ ~ M-f\ L 0 N1"\ LLrtv1() ";: Ljt00 @ rv-.L GIVEN: AB =:::.EB,FB~ eB - L OMN ~~1 @ NM:MN ~~S\L.G> M L tv\ (\) J ;:j'\L (VI f'J O"t '" L DtvJ Cu'i\M w-. ~\\.A;') LQSP~LQS¥- CD @ 4. NtJ\L; '('IlL (\)N\O -4f;\LOML @ L M f'J J ~ L. N \V\ l (J) A MJN ~ 6fV tAv\ GIVEN: RS =:::. S\I'k)'lrlv1iv- AS 1\_ >-I. CL1t'j'" ST, PU=:::.PQ - PROVE: BG=:::.BD LUPT=:::.LQPR A PROVE: 8.PTU =:::. !:::.PR Q p c F u fB (j) rw;~ @ .f-{I;~ m ~ LJ\g'f~L E6(., (!j) 6~f~ tJ E&.. ~LGAB~ @ OJ Lb£B 4 GABr:;~ D&6 fJ(;;, ~ 'BD Q E ~,,"\-tn ~~~.~~ (\~h ~Uhr,.u.... c~~j'''''J r'''rts IDA 'fW,UJ C;J{\ Lm-f.,-JrlfrlJ,~r,,-;f> CD @) rU~ Fa @ LUPT~LGff2 @) p£ ~ fS @ bPS~6f2>{2. ~1V¥ '~~ 'j'v.t1' ~W"t.d..)~ ~~,,,Ji.., SkSOi~ @ Pf~PfL o.¥yt'f.,.l1~pI-vis (j) b.PT\J~ 6P~ b~~ / .• -f'-v<--