4.3 Proving Triangles Are Congruent: SSS And SAS
Page 1 of 84.3Proving Triangles areCongruent: SSS and SASWhat you should learnGOAL 1 Prove thattriangles are congruent usingthe SSS and SASCongruence Postulates.GOAL 1SSS AND SAS CONGRUENCE POSTULATESHow much do you need to know about two triangles to prove that they arecongruent? In Lesson 4.2, you learned that if all six pairs of corresponding parts(sides and angles) are congruent, then the triangles are congruent.BUse congruencepostulates in real-lifeproblems, such as bracing astructure in Example 5.GOAL 2FACWhy you should learn itIf Congruence postulateshelp you see why trianglesmake things stable, such asthe seaplane’s wing belowand the objects inExs. 30 and 31.AL LISides arecongruentAngles arecongruentandÆÆÆÆÆÆ1. AB DEDTriangles arecongruentthen4. A D2. BC EF ABC DEF5. B E3. AC DF6. C FFEREEIn this lesson and the next, you will learn that you do not need all six of thepieces of information above to prove that the triangles are congruent. Forexample, if all three pairs of corresponding sides are congruent, then theSSS Congruence Postulate guarantees that the triangles are congruent.P O S T U L AT ESide-Side-Side (SSS) Congruence PostulatePOSTULATE 19If three sides of one triangle are congruent to three sides of a secondtriangle, then the two triangles are congruent.ÆIfMÆSideMN QR ,ÆÆSideNP RS , andÆÆSidePM SQ ,then MNP QRS.EXAMPLE 1qPNSRUsing the SSS Congruence PostulateProve that PQW TSW.PTParagraph Proof The marks on the diagramÆÆ ÆÆÆÆshow that PQ TS, PW TW, and QW SW. 212qWSo, by the SSS Congruence Postulate, you know that PQW TSW.Chapter 4 Congruent TrianglesS
Page 2 of 8ACTIVITYConstructionCopying a TriangleCFollow the steps below to construct a trianglethat is congruent to a given ABC.ABFD1DEÆConstruct DE so thatÆit is congruent to AB.(See page 104 for theconstruction.)2FDEOpen your compassto the length AC. Usethis length to draw anarc with the compasspoint at D.3EDDraw an arc withradius BC and center Ethat intersects the arcfrom Step 2. Label theintersection point F.EDraw DEF. By theSSS CongruencePostulate, ABC DEF.4The SSS Congruence Postulate is a shortcut for proving two triangles arecongruent without using all six pairs of corresponding parts. The postulate belowis a shortcut that uses two sides and the angle that is included between the sides.STUDENT HELPStudy TipIn the triangle, B is theincluded angle betweenÆÆsides AB and BC .CAP O S T U L AT EIf two sides and the included angle of one triangle are congruent to twosides and the included angle of a second triangle, then the two trianglesare congruent.ÆIfBSide-Angle-Side (SAS) Congruence PostulatePOSTULATE 20ÆqSidePQ WX ,Angle Q X, andÆÆSideQS XY ,then PQS WXY.EXAMPLE 2XSPWUsing the SAS Congruence PostulateProve that AEB DEC.CB1EAStatementsÆY2DReasonsÆ ÆÆ1. AE DE, BE CE1. Given2. 1 22. Vertical Angles Theorem3. AEB DEC3. SAS Congruence Postulate4.3 Proving Triangles are Congruent: SSS and SAS213
Page 3 of 8MODELING A REAL-LIFE SITUATIONGOAL 2Choosing Which Congruence Postulate to UseEXAMPLE 3LogicalReasoningDecide whether enough information is given in the diagram to prove that PQR PSR. If there is enough information, state the congruence postulateyou would use.qSPRSOLUTIONÆÆParagraph Proof The marks on the diagram show that PQ PS andÆÆÆÆQR SR. By the Reflexive Property of Congruence, RP RP. Because thesides of PQR are congruent to the corresponding sides of PSR, you can usethe SSS Congruence Postulate to prove that the triangles are congruent.Proving Triangles CongruentEXAMPLE 4REFELAL IARCHITECTURE You areDdesigning the window shownin the photo. You want to make DRAcongruent to DRG. You design theÆÆÆÆwindow so that DR fi AG and RA RG.Can you conclude that DRA DRG?ARGSOLUTIONTo begin, copy the diagram and label itusing the given information. Then writethe given information and the statementyou need to prove.ProofÆÆÆÆGIVEN DR fi AG ,DARA RGRPROVE DRA DRGStatementsÆReasons1. DR fi AG1. Given2. DRA and DRG are2. If 2 lines are fi, then they form 4 rt. .3.4.5.6.214Æright angles. DRA DRGÆÆRA RGÆÆDR DR DRA DRGChapter 4 Congruent Triangles3. Right Angle Congruence Theorem4. Given5. Reflexive Property of Congruence6. SAS Congruence PostulateG
Page 4 of 8EXAMPLE 5REFELAL ITriangular Frameworks are RigidSTRUCTURAL SUPPORT To prevent a doorway from collapsing after anearthquake, you can reinforce it. Explain why the doorway with thediagonal brace is more stable, while the one without the brace can collapse.SOLUTIONIn the doorway with the diagonal brace, the wood forms triangles whose sideshave fixed lengths. The SSS Congruence Postulate guarantees that these trianglesare rigid, because a triangle with given side lengths has only one possible sizeand shape. The doorway without the brace is unstable because there are manypossible shapes for a four-sided figure with the given side lengths.xyUsingAlgebraEXAMPLE 6Congruent Triangles in a Coordinate PlaneUse the SSS Congruence Postulateto show that ABC FGH.A ( 7, 5)yC ( 4, 5)B ( 7, 0)1H (6, 5)G (1, 2)F (6, 2)1xSOLUTIONÆÆBecause AC 3 and FH 3, AC FH. Because AB 5 and FG 5,ÆÆAB FG. Use the Distance Formula to find the lengths BC and GH.22 x ( y2 º y d (x2º1) 1) STUDENT HELPLook BackFor help with theDistance Formula,see page 19.22d (x x ( y2 º y 2º1) 1) GH (6 º 1 )2 (5 º 2 )22BC 4 7 2 5 0 3 2 52 5 2 32 3 4 3 4 ÆÆBecause BC 3 4 and GH 3 4 , BC GH. All three pairs ofcorresponding sides are congruent, so ABC FGH by theSSS Congruence Postulate.4.3 Proving Triangles are Congruent: SSS and SAS215
Page 5 of 8GUIDED PRACTICEVocabulary Check 1. Sketch a triangle and label its vertices. Name two sides and the includedangle between the sides.Concept Check 2. ERROR ANALYSIS Henry believes he can use the information given in thediagram and the SAS Congruence Postulate to prove the two triangles arecongruent. Explain Henry’s mistake.Skill Check LOGICAL REASONING Decide whether enough information is given toprove that the triangles are congruent. If there is enough information, tellwhich congruence postulate you would use.3. ABC, DEC4. FGH, JKHAB5. PQR, SRQqPFGCEHDKSRJPRACTICE AND APPLICATIONSSTUDENT HELPExtra Practiceto help you masterskills is on p. 809.NAMING SIDES AND INCLUDED ANGLES Use the diagram. Name theincluded angle between the pair of sides given.ÆÆÆÆÆÆ6. JK and KLÆÆÆÆJ7. PK and LK8. LP and LKL9. JL and JKÆ10. KL and JLÆK11. KP and PLPLOGICAL REASONING Decide whether enough information is given toprove that the triangles are congruent. If there is enough information, statethe congruence postulate you would use.12. UVT, WVT13. LMN, TNMLT14. YZW, YXWTZWYSTUDENT HELPWVMUNXHOMEWORK HELPExample 1: Exs. 18,20–28Example 2: Exs. 19–28Example 3: Exs. 12–17Example 4: Exs. 20–28Example 5: Exs. 30, 31Example 6: Exs. 33–3515. ACB, ECDAT17. GJH, HLKUGJLCB21616. RST, WVUDChapter 4 Congruent TrianglesERSVWHMK
Page 6 of 8DEVELOPING PROOF In Exercises 18 and 19, use the photoÆÆÆÆof the Navajo rug. Assume that BC DE and AC CE .18. What other piece of informationis needed to prove that ABC CDE using theSSS Congruence Postulate?19. What other piece of informationis needed to prove that ABC CDE using theSAS Congruence Postulate?20.DEVELOPING PROOF Complete the proof by supplying the reasons.ÆÆÆÆGIVEN EF GH ,HEGFFG HEPROVE EFG GHEStatementsÆÆÆÆÆÆReasons? 1. ? 2. 1. EF GH2. FG HE? 3. ? 4. 3. GE GE4. EFG GHETWO-COLUMN PROOF Write a two-column proof.ÆÆÆÆÆÆ21. GIVEN NP QN RS TR,PQ STPROVE NPQ RSTNSÆÆ ÆÆ22. GIVEN AB CD, AB CDPROVE ABC CDAATD12qPINTSTUDENT HELPNEER THOMEWORK HELPVisit our Web sitewww.mcdougallittell.comfor help with paragraphproofs.RBCPARAGRAPH PROOF Write a paragraph proof.Æ 23. GIVEN PQ bisects SPT,ÆÆSP TPPROVE SPQ TPQÆÆ ÆPROVE PQT RSTqPPSTTqÆ24. GIVEN PT RT, QT STSR4.3 Proving Triangles are Congruent: SSS and SAS217
Page 7 of 8PROOF Write a two-column proof or a paragraph proof.ÆÆÆ25. GIVEN AC BC,Æ ÆÆ26. GIVEN BC AE, BD AD,ÆÆM is the midpoint of AB.ÆDE DCPROVE ACM BCMPROVE ABC BAECBCDAMBÆÆÆÆEAÆ27. GIVEN PA PB PC,ÆÆ ÆÆÆÆ28. GIVEN CR CS, QC fi CR,AB BCPROVE PAB PBCQC fi CSPROVE QCR QCSqPCCINTSTUDENT HELPNEER T29.SOFTWARE HELPVisit our Web sitewww.mcdougallittell.comto see instructions forseveral softwareapplications.RBASTECHNOLOGY Use geometry software to draw a triangle. Draw a lineand reflect the triangle across the line. Measure the sides and the anglesof the new triangle and tell whether it is congruent to the original one.Writing Explain how triangles are used in the object shown to make itmore stable.30.32.31.CONSTRUCTION Draw an isosceles triangle with vertices A, B, and C.Use a compass and straightedge to construct DEF so that DEF ABC.xy USING ALGEBRA Use the Distance Formula and the SSS CongruencePostulate to show that ABC DEF.33.34.yCEDy35.FyEDDCAE21ABF5 xC1B1218Chapter 4 Congruent TrianglesFxAB1x
Page 8 of 8TestPreparationÆÆ ÆÆ36. MULTIPLE CHOICE In RST and ABC, RS AB, ST BC, andÆÆTR CA. Which angle is congruent to T?A¡B¡ RC¡ AD¡ Ccannot be determined37. MULTIPLE CHOICE In equilateral DEF, a segment is drawn from point FÆto G, the midpoint of DE. Which of the statements below is not true? ChallengeA¡ÆB¡ÆDF EFÆÆDG DFC¡ÆÆDG EGD¡38. CHOOSING A METHOD Describe howyto show that PMO PMN using theSSS Congruence Postulate. Then find a wayto show that the triangles are congruent usingthe SAS Congruence Postulate. You may notuse a protractor to measure any angles. Comparethe two methods. Which do you prefer? Why?EXTRA CHALLENGEwww.mcdougallittell.com DFG EFGNM1O1xPMIXED REVIEWSCIENCE CONNECTION Find an important angle in the photo. Copy the angle,extend its sides, and use a protractor to measure it to the nearest degree.(Review 1.4)39.40.USING PARALLEL LINES Find m 1 and m 2. Explain your reasoning.(Review 3.3 for 4.4)41.42.43.11 129 212257 LINE RELATIONSHIPS Find the slope of each line. Identify any parallel orperpendicular lines. (Review 3.7)44.45.yAED1xCyG112B46.yFœxHRP 114.3 Proving Triangles are Congruent: SSS and SASx219
Aug 27, 2015 · 4.3 Proving Triangles are Congruent: SSS and SAS 213 The SSS Congruence Postulate is a shortcut for proving two triangles are congruent without using all six pairs of corresponding parts. The postulate below is a shortcut that uses two sides and the angle that is includedbetween the sides. Using the SAS Congruence Postulate Prove that AEB .
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Similar Triangles Solve Problems Using Properties of Similar Triangles Prove Similar Triangles · Use Definitions, Postulates, Theorems to Prove Triangles are Similar · Use Algebraic and Coordinates Methods to Prove Triangles are Similar Prove Triangles are Similar Using Deductive Proofs G.8 Right Triangles Pythagorean Theorem
Feb 14, 2011 · Ch 4: Congruent Triangles 4‐1 Congruent Figures 4‐2 Triangle Congruence by SSS and SAS 4‐3 Triangle Congruence by ASA and AAS 4‐4 Using Congruent Triangles: CPCTC 4‐5 Isosceles and Equilateral Triangles 4‐6 Congruence in Right Triangles 4‐7 Using Corresponding Parts of Congruent Triangles
Jul 05, 2018 · Congruent Triangles Reasoning and Proof Reasoning and proof of congruent triangles involves proving congruence of at least two triangles according to five congruent triangles theorems (Secondary School Mathematics Section by People’s Education Press, 2013). The five congruent triangles theorems have the following types: 1.
and SSS similarity statements. . .And Why To measure height indirectly, as in Example 4 In this lesson, you will show triangles are similar without using the definition of similar triangles.The two triangles shown above suggest the following postulate. 7-3 11 The AA Postulate and the SAS and SSS Theorems Activity: Triangles with Two Pairs of .
Congruent Triangles Strand: Triangles Topic: Exploring congruent triangles, using constructions, proofs, and coordinate methods Primary SOL: G.6 The student, given information in the form of a figure or statement will prove two triangles are congruent. Related SOL: G.4, G.5 Materials Congruent Triangles: Shortcuts activity sheet (attached)
a) Name the similar triangles. b) Write an extended proportion that is true for these triangles. c) If t 2, a 3, and n 6, find e. 6. Consider the triangles shown below. G R E T A a) Name the similar triangles. b) Write an extended proportion that is true for these triangles. c) If g 30, AT 36, and t 45, find GR. 7. Name the similar .
You need to check your peak flow: every day, twice a day to get a useful pattern of scores at the same times of day, in the morning and in the evening before you take your asthma medicine otherwise it will change the score using your best effort each time you blow into the meter so you’re comparing like with like using the same peak flow meter each time.
