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NAME _ DATE Practice with Examples For use with pages 194-201 Classify triangles by their sides and angles and find angie measures in triangles A triangle is a figure formed by three segments joining three noncollinear points. An equilateral triangle has three congruent sides. An isosceles triangle has at least two congruent sides. A scalene triangle has no congruent sides. An acute triangle has three acute angles. An equiangular triangle has three congruent angles. A right triangle has one right angle. I An obtuse triangle has one obtuse angle. . II The three angles of a triangle are the interior angles. I When the sides of a triangle are extended, the angles that are adjacent to the interior angles are exterior angles. I Theorem 4.1 Triangle Sum Theorem The sum of the measures of the interior angles of a triangle is 180°. I I Theorem 4.2 Exterior Angle Theorem , The measure of an exterior angle of a triangle is equal to the sum of the measures of the two nonadjacent interior angles. Corollary to the Triangle Sum Theorem The acute angles of a right triangle are complementary. Copyright © McDougal Littell Inc. All rights reserved. Practice Workbook Geometll"Y vvit:h Examples I -, LESSON NAME _ \ DATE I Practice with Examples CONTINUED for use with pages 194-201 ~ C/ass,_ifl.:;..y'_"n.;;.9_Tf:_iB_fl..::;:y_16_S """""""' _ Classify the triangles by their sides and angles. a. b. L z 45° 5.7 4 J~";;";"----'-~ 16 45° XL.......J....----....:..::..~y 4 SOLUTiON a. D.JKL has one obtuse angle and no congruent sides. It is an obtuse scalene triangle. b. LlXYZ has one right angle and two congruent sides. It is a fight isosceles triangle . .~~.'!!.~~~i!.~ ..~~.~. ~l!.i!.I!!.I!.~'!..! '" . Classify the triangie by its sides and angles. 1. 2. 5 3. 5.8 Geomet!'\f Practice Workbook with Examples 18 Copyright © McDougal Littell Inc . .All rights reserved. LESSON NAME -------------------- DATE Practice IIvith Examples CONTINUED For use with pages 194-201 MUBIJ.6'~ Measures b. Find the value of y. a. Find the value of x. (4x - 5)° 2VO I (3x+ 11)" 50° f I I a. From the Corollary to the Triangle Sum Theorem, you can write and solve an equation to find the value of x. The acute angles of a right triangle are complementary. I x "I i = 12 Solve for x. b. You can apply the Exterior Angle Theorem to write and solve an equation that will allow you to find the value of y. 90° + 50° = 2yo Apply the Exterior Angle Theorem. y = 70 Solve for y. !..;:.~!.~~~i!.~.!.~.~.~lJ.?l!!l!.~i!..? Find the value of 4. Copyright © McDougal Littell Inc. All rights reserved. . x. 5. Geometry Practice Workbook with Examples NAME DATE ----------------------------------~------ Practice with Examples For use with pages 202-210 identify congruent figures and corresponding parts VOCABULARY When two geometric figures are congruent, there is a correspondence: between their angles and sides such that corresponding angles are congruent and corresponding sides are congruent. Theorem 4.3 Third Angles Theorem If two angles of one triangle are congruent to two angles of another triangle, then the third. angles are also congruent. _ Using Properties of Congruent Figures In the diagram, ABCDE == FGHIJ a. Find the value of x. b. Find the value of y. B F (3x+ 4) A J D G ; E J SOLUTiON a. You know that AE SoAE 10 = = 3x == Fl. b. You know that LD Fl. So, mLD + 4r = (8y - 9)° 56 = 8y 4 x=2 = == LI. mLI. y=7 ) Geometry Practice Workbook with Examples Copyright © McDougal Littell Inc. All rights reserved. LESSON NAME _ DATE Practice with Examples CONTINUED For use with pages 202-210 Exercises for Example 1 ....................................................................................................................................... In Exercises 1 and 2, for each pair of figures find (a) the value of x and (0) the value of y. 1. MBC-=MEF B F C A (5y - 3)° 2. ABCDEF -= GHIJKL 0 G (4x - 5)" L A E j' ). F K I I Using the Third Angles Theorem Find the value of x. c o F A .••....•.. ~ __ f..LJ E B SOLUTiON In the diagram, LA = LD and LB = LE. From the Third Angles Theorem, you know that LC LF. So, mLC = mLF. -= From the Triangle Sum Theorem, mLC = 180° - 30° - 110° ~ 40°. lI mLC = mLF 40 = x J! .Third Angles Theorem Substitute. t I I , -! I ) Copyright © McDougal Littell Inc. All rights reserved. Geometry Practice Workbook with Examples LESSON NAME CONTINUED ._· _ ~--- DATE· _ : ""'." Practice with Examples For use with pages 202-210 .~~.~!.I?!.~f!.~. !.~L ~1!.~I!!.I!.~f!. ~ ; Find the value of x. 3. 4. Geometry Practice Workbook with Examples Copyright © McDougal Littell Inc. All rights reserved. NAME _ DATE Practice with Examples For use with pages 212-219 .......- . - =l Prove that triangles are congruent using the SSS and SAS Congruence Postulates Postulate 19 Side-Side-Side (SSS) Congruence Postulate . If three sides of one triangle are congruent to three sides of a second triangle, then the two triangles are congruent. . I • Postulate 20 Side-Angle-Side (SAS) Congruence Postulate I. .,EI» ~ If two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then the two triangles are congruent. . Using the SAS Congruence Postulate Prove that !:ABC == 6DEF. B F o c E A S«:U.UTRON The marks on the diagram show that AB == DE, BC == EF, and LB == LE. So, by the SAS Congruence Postulate, you know that 6ABC~ 6DEF . .~~~!.~~~~.~. !.~.~ .~1!.~I!!p./f!..! . State the congruence postulate you would use to prove that the two triangles are congruent. 1.. Copyright © McDougal Littetllnc. All rights reserved. 2. Georne1:ry Practice Workbook with Examples 2 LIESSON NAME DATE ~----:-_---, Practice with Examples CONTINUED For use with pages 212-219 ~.'~-. WJ< ~. Congruent Triangles in a Coordinate Plane. Use the SSS Congruence Postuiate to show that MBC == 6CDE. SOLUTION . Use the distance formula to show that corresponding sides are the same length. For all lengths, d = .)(x2 - XI)2 + (h - Yl)2· AB = .)(-3- (-4))2 + (-3 - 2)2 = .) CD = 12 + (- 5)2 = .J52 + = .J26 = = . BC .J( -1 - (- 3))2 + (0 - (- 3))2 = .J22+ 32 = CA =. DE, = .J( -:-4 -' (-1))2 = .J(-3)2 + 22 = JI3 So, CA = (1 - 0)2 12 . .J26 ,--; DE = .J(1 - 4)2 + (3 - 1)2 = and hence + =.J(-3)2+22 JI3 So, BC (-1))2 AB == ro. So, AB = CD, and hence .J(4- BE == DE. + (2 - 0)2 EC, and hence CA ffi EC = .)(-1 - 1)2 + (0 - 3)2 = .J(-2)2 + (-3)2 = .JI3 == Ee. So, by the SSS Congruence Postulate, you know that !::,ABC == 6CDE. Geometry Practice Workbook vvith Examples Copyright © McDougal Littell Inc. All rights reserved. LESSON CONTINUED NAME ~ _ DATE Practice with Examples For use with pages 212-219 Exercise for .,,"Example ...................•.... 2 ......... 3. Prove that MBC == 6.DEF. ~""""""""""" "" "" "" "" , " " " " " " ". " "" '"'' "'" "" . '; ~ ' Copyright © McDougal Littell Inc. All rights reserved. Geometry Practice Workbook with Examples, 9 t I NAME DATE ~----------~---------------------- \ Practice with Examples \ \ For use with pages 220-227 1 Prove that triangles are congruent using the ASA Congruence Postulate and the AAS Congruence Theorem . , , I t 1 Postulate 21 Angle-Side-Angle (ASA)'Congruence Postulate If two angles and the included side of one triangle are congruent to two angles and the included side of a second triangle, then the two triangles are congruent. , I Theorem 4.5 Angle-Angle-Side (AAS) Congruence Theorem If two angles and a nonincluded side of one triangle are congruent to two ! angles and the corresponding nonincluded side of a second triangle, then the two triangles are congruent. I _ Proving Triangles are Congruent Using the ASA Congruence Postulate ~ ~ == EC, LB == !::ABC == 6DEC Given: BC Prove: LE = 8 Statements Reasons == Ee 1. Given 2. LB== LE 2.. Given == !::ABC == 3. LACB LDCE 3. Vertical Angles Theorem 4. 6DEC 4. ASA Congruence Postulate Geometry Practice Workbook ms o SOLUTION 1. Be em with Examples i Copyright © McDougal Littell Inc. All rights reserved. --~ LESSON NAME ~ · D~E Practice with Examples -"""'::r, CONTINUED :..1;;.. ',. _"--4f For use with pages 220-227 Exercises lor Example t .............................................................................................................................................. In Exercises 1 and 2, use the given information the triangles are congruent. 1. Given: MC == AC LNMC to prove that == DE, LA == Prove: 6BAE == 6CDE 2. Given: AE and LBAC are right angles. Prove: 6NMC == 6BAC N o "'-'--------" C B ~ Proving Triangles ere Congruent Using AAS Congruence Theorem !Sf W~ ~ . Given: AD == AE, LB == L C Prove: MBD == ~wnr lZi.IIlII&\!:WZ ::::a:SWt::::=i:i :::eu!'!'JW~.= c B MCE A Statements Reasons 1. AD==AE 2. LB 3. LA == == LC 2. Given LA 3. Reflexive Property of Congruence 4. MBD==MCE Copyright © McDougal Littell Inc. All rights reserved. . 1. Given 4. AAS Congruence Theorem Geometry Practice Workbook with Examples LD LESSON· 4.4 CONTINUED NAME __'~~~~~~~~~~~~~~~~~~~~~ __ DATE ~ _ Practice with Examples For use with pages 220-227 ~~.f!!.~~~f!.~. !.C!!. .~l!.~I!!p/f!..? . In Exercises 3 and 4, use the given information the triangles are congruent, 3. Given: LG == LB, CB II GA to prove that 4. Given: LOMN == LONM, LLMO== LINO Prove: ·6GCA == 6BAC Prove: 6MIN == 6NLM 'M A"-----+--\"78 L G J Geometry Practice Workbook vvith Examples Copyright © McDougal Littell Inc. All rights reserved. NAME ~ _ DATE Practice with Examples For use with pages 229-235 Use congruent triangles to plan and write proofs == PQ, QS == RT Given: PR Prove: SR == TQ p Plan for Proof: QS and RT are corresponding parts of 6PQS and 6PRT and also of 6RQS and 6QRT. The first set of triangles is easier to prove congruent than the second set. Then use the fact that corresponding parts of congruent triangles are congruent. R Q SOU..!ITION Statements 1. PR . Reasons == PQ 1. Given 2. PR = PQ 3. PR = PS 4. PQ = PT 6. SR + SR == TQ 7. SR = 5. PS 8. PS 2. Definition of congruence + SR + TQ = PT TQ 11. 6PQS 12. QS Copyright © McDougal Littell Inc. All rights reserved. == RT TQ 5. Subsitution 8. Subtraction property of equality 9. Definition of congruence LP == + 7. Definition of congruence = PT == 4. Segment Addition Postulate 6. Given 9. PS== PT 10. LP 3. Segment Addition Postulate 10. Reflexive Property of Congruence 6PRT 11. SAS Congruence Postulate 12. Corresponding parts of congruent triangles are congruent. . Practice Workbook Geometry with Examples LESSON NAME _ DATE Practice with Examples CONTINUED For use with pages 229-235 Exercises for Example 1 ....................................................................................................................................... . . Use the given information to prove the desired statement. 1. Given: RT==AS, RS==AT Prove: L TSA 2. Given: == LSTR Prove: Ll==L2==L3,L4==L5,ES==DT HE == HD H R A E T It>?} o ,"J s \ ~ , Using More than One Pair of Triangles Given: Ll == L2, L5 == L6 B Prove: AC.lBD Plan for Proof: It can be helpful to reason backward from what is to be proved. You can show that AC.lBD if you can show L3 == L4. Notice that L3 and L4 are corresponding parts of MBO and MbO. You can prove MBO == bADO by SAS if you first prove AB == AD. AB and AD are. corresponding parts of MBC and MDC. You can prove MBC == MDC by ASA. A~----::------':-t-"---=---7C o ./.; lGeometry Practice Workbook with Examples Copyright © McDougal Littell Inc. All rights reserved. LESSON NAME _ DATE Practice with Examples CONTINUED For use with pages 229-235 Statements 1. Ll ~ L2,L5 Reasons ~ L6 1. Given 2. Ae~AC 2. Reflexive Property of Congruence 3. MBe~MDe 3. ASA Congruence Postulate 4. AB~AD 4. Corresponding parts of congruent triangles are congruent. 5. AO ~AO 5. Reflexive Property of Congruence 6. MBO==MDO 6. SAS Congruence Postulate . 7. L3 ~ L4 7. Corresponding parts of congruent triangles are congruent. 8. If 2 lines intersect to form a linear pair of congruent angles, then the lines are 1.. ~;:.~!.'?/~~~. t~.~ .~l!:?'!!l!.~~.? . In Exercises :3and 4, use the given information desired statement. to prove the 4. Given: LDAL ~ LBCM, 3. Given: PA ~ KA, LA ~ NA Prove: AX ~ AY LeDL~ LABM DC~BA N Prove: AL == eM rc+r+: L Copvriqht © McDougal Littstl lnc, All rights reserved, A -1- -; C B Geometry Practice Workbook with Examples NAME __ ~ ~ _ DATE .Practice with Examples For use with pages 236-242 Use properties OIf isosceles, equilateral, and right triangles I VOCABULARY If an isosceles triangle has exactly two congruent sides, the two angles adjacent to the base are base angles. I I If an isosceles triangle has exactly two congruent sides, the angle opposite the base is the vertex angle. I Theorem I I. 4.6 Base Angles Theorem If two sides of a triangle are congruent, then the angles opposite them are congruent Theorem 4.7 Converse of the Base Angles Theorem If two angles of a triangle are congruent, then the sides opposite them I are congruent. 1i~ Corollary to Theorem 4.6 If a triangle is equilateral, then it is equiangular. Corollary to Theorem 4.7 , If a triangle is equiangular, then it is equilateral. Theorem 4.8 Hypotenuse-Leg (HL) Congruence Theorem If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and a leg of a second right triangle, then the two triangles are congruent. Using Properties.of Right Triangles ~ I~ -, Given that LA and LD are Tight angles and AB == DC, show that MBC == l',.DCB. - A~ ~ __ ~C SC)l.UTION Paragraph proof You are given thatLA and LD are right angles. By definition, MBC and l',.DCB are right triangles. You are also given that a leg of MBC, AB, is congruent to a leg of ~DCB, DC. You know that the B D hypotenuses of these two triangles, BC for both triangles, are congruent because BC == BC by the Reflexive Property of Congruence. Thus, by the Hypotenuse-Leg Congruence Theorem, MBC == /'::,DCB. Geomet!'y Practice Workbook with Examples Copyright © McDougal Littell Inc. All rights reserved. LESSON NAME ~ _ DATE Practice with Examples l ~- CONTINUED For use with pages 236-242 .~~~!.~~~f!.~. !.~.~.~lf.f!.I!!.e.~l!.. ! . Write a paragraph proof. 1. Given: BCl.AD, Prove: AB == DB = 2. Given:mLIKL MBC == L.DBC IL . ! Prove: B mLMLK == MK IK==ML J ~ A C 0 L Using Equilateral and Is,!sceles.., T;iangles M . , Find the values of x and y. SOLUTiON Notice that tiABC is an equilateral triangle. By the Corollary to Theorem 4.6, /':,.ABCis also an equiangular triangle. Thus mLA = mLABC = mLACB = 60°. So, x = 60. A Notice also that f:..DBC is an isosceles triangle, and thus by the Base Angles Theorem, mLDBC = mLDCB. Now, since mLABC = mLABD + mLDBC, mLDBC = 60° - 30° = 300. Thus, y = 30 by substitution. Copyright © McDougal Littell Inc. All rights reserved. Geometry Practice Workbook with Examples = 90°, LESSON NAME CONTINUED ~ ~----------------~-- _ DATE Practice with Examples « For use with pages 236-242 ~ Exercises for Example 2 - ........................................................................................................................................ Find the values of x and y. 3. 4. 20 5. 7 Geometry Practice Workbook with Examples Copyright © McDougal Littell Inc. All rights reserved. . NAME ~ __ ~-------- DATE Practice with Examples For use with pages 243-250 Place geometric figures in a coordinate plane and write a coordinate proof VOCABULARY I A coordinate proof involves placing geometric figures in a coordinate I plane and then using the Distance Formula and the Midpoint Formula, I as well as postulates and theorems, to prove statements about the figures. I t _ ., Using the Distance formula A right triangle has legs of 6 units and 8 units. Place the triangle in a coordinate plane. Label the coordinates of the vertices and find the length of the hypotenuse. One possible placement of the triangle is shown. Points on the same vertical segment have the same x-coordinate, and points on the same horizontal segment have the same y-coordinate. y I I ! ! r ! ! ! Distance Formula 1 Substitute. :3 i i = -fiOo = 10 - i (O,O)! I ! i I ! ! I I I ! 1(8,0) I I I I I _ Use a coordinate plane and the Distance Formula to answer the question. ' 1. A rectangle has sides of length 8 units and 2 units. What is the length of one of the diagonals? Copyright © McDougal Littell Inc. I I -1 xl i Simplify and evaluate square root. .~~.f!!.l?,~~~.~. !.C!.~.~~c:.'!!P..~f! ..! All rights reserved. I I i(8,6) I I I I Y i ! i VI I i dlA I 1/ l/, AI ! = --1(8 - 0)2 + (6 - 0)2 , /' Use the Distance Formula to find d. .\- i I I ! Geometry Practice Workbook vvith Examples . LESSON NAME _ DATE CONTINUED For use with pages 243-250 In the diagram, 6ABD == 6CBD. y Find the coordinates of point B. SOLUY§ON . Because the triangles are congruent, AB == CB. So, point B is the midpoint of AC: This means you can use the Midpoint Formula to find the coordinates of point B. ) _ (Xl + B(x, )) - X') 2' )'1 + 2 Y2) ~-(-72+ 0, ° 27' ) -_ (-22,22) T Midpoint Formula Substitute and Simplify . .~;:.~!f?~~f!.~.!.l!.~.~l!.l!.i!!.I?~f!..? 2. In the diagram, 6FGH coordinates of H. Geometry Practice Workbook with Examples == . 6JIH. Find the Copyright © McDougal Littell Inc. All rights reserved. LESSON NAME ~ ~ _ DATE Practice with Examples CONTINUED for use with pages 243-250 WO'itillll1lI a Coordinate Write a plan to prove that 6,DEF == 6,DGF. y E(b, c) Given: Coordinates of figure DEFG. Prove: 6DEF == 6DGF 'F(a,O) x Plan for Proof Use the Distance Formula to show that segments EF and GF have equal lengths and that segments DE and DG have equal lengths. Use the Reflexive Property of Congruence to show that DF 2:: DF. Then use SSS Congruence Postulate to conclude that 6DEF 2:: 6DGF. Gib, -c) .~~.~~~~~l!.~. !.~.~ .~l!.C?r!!.I!.~~.? . Describe a pian for the proof. y 3. Given: Coordinates of figure ABeD. Prove: MBC == 6CDA A(O,O) ~~------~--------~~ x D(2a, -b) ,I Copyright © McDougal Littell Inc. All rights reserved. Geometry Practice Workbook with. Examples