Triangle Congruence 2 4 By SSS And SAS - IMater

Mathematics Florida StandardsTriangle Congruenceby SSS and SAS42MAFS.912.G-SRT.2.5 Use congruence. criteria fortriangles to solve problems and prove relationships Ingeometric figures.MP1,MP 3, MP4,MP7JObjective To prove two triangles congruent using the SSS and SAS PostulatesGetting Ready! ' XSfAre the triangles below congruent? How do you know?1!i c 1-8 L—1-A0How con you tell-p.whether theseoA1i!11m1M—j11Mi li,1of information15required to tell iftwo triangles are12,1L.k JP11Zthe least amountM—11 -A.triangles arecongruent? In thislesson, you will learnt1L 11I 8 ' 10lD \x12 1Li4 Li.6 ii i "IM1 ' ! 1 1congruent.MATHEMATICALPRACTICESIn the Solve It, you looked for relationships between corresponding sides andangles. In Lesson 4-1, you learned that if two triangles have three pairs of congruentcorresponding angles and three pairs of congruent corresponding sides, then thetriangles are congruent.If you know./-F ufg 7k H /LLFH 1L.then you know AFGH AJKL.However,this is more information about the corresponding parts than you need toprove triangles congruent.Essential Understanding You can prove that two triangles are congruentwithout having to show that all corresponding parts are congruent.In this lesson,you will prove triangles congruent by using(1)three pairs of corresponding sidesand(2)two pairs of corresponding sides and one pair of corresponding angles.226Chapter 4 Congruent TrianglesC

Postulate 4-1Side-Side-Side (SSS) PostulatePostulateIf.Then .If the three sides of oneAB W, ,AC AABC s ADEFtriangle are congruent toBEthe three sides of anothertriangle, then the twotriangles are congruent.As described in Chapter 1, a postulate is an accepted statement offact. The Side-SideSide Postulate is perhaps the most logical fact about triangles. It agrees with the notionthat triangles are rigid figures; their shape does not change until pressure on their sidesforces them to break. This rigidity property is important to architects and engineerswhen they build things such as bicycle frames and steel bridges.Using SSSF".You have two pairsof congruent sides.What else do youGiven: LM NP, IP NMProve: ALMN - ANPLneed?You need a third pair ofcongruent correspondingsides. Notice thatthe triangles share acommon side, W.LM NPGivenLN LNReflexive Prop, ofLP NMGivenALMN ANPLSSSGof If? 1. Given: BC BF, CD FDDProve: ABCD ABFDCPowerGeometiy.comLesson 4-2 Triangle Congruence by SSS and SAS227

You can also show relationshipsbetween a pair of correspondingLA is indudedsides and an included angle.between BABC is includedbetween LBand lC.and AC.Ihe word included refers to theangles and the sides of a triangleas shown at the right.Postulate 4-2 Side-Angle-Side (SAS) PostulatePostulateIf.Then .If two sides and theAB ' ,LA LD,AABC ADEFincluded angle of onetriangle are congruent toAC Wtwo sides and the includedangle of another triangle,then the two triangles arecongruent.You likely have used the properties of the Side-Angle-Side Postulate before. Forexample, SAS can help you determine whether a box will fit through a doorway.Suppose you keep your arms at a fixed angle as you move from the box to the doorway.Ihe triangle you form with the box is congruent to the triangle you form with thedoorway.Ihe two triangles are congruent because two sides and the included angle ofone triangle are congruent to the two sides and the included angle of the other triangle.228Chapter 4 Congruent Triangles

Problem 2IDo you need anotherpair of congruentUsing SASWhat other information do you need to proveADEF AFGD by SAS? Explain.sides?Look at the diagram.The triangles share Of.So, you already have twopairs of congruent sides.The diagram shows that EF GD.Also, DF DF bythe Reflexive Property of Congruence. To prove thatADEF AFGD by SAS, you must have congruentincluded angles. You need to know that AEFD AGDF.Got It? 2. What other information do you need to proveALEB ABNLbySAS?Recall that, in Lesson 1-6, you learned to constructsegments using a compass open to a fixed angle. Nowyou can show that it works. Similar to the situation withthe box and the doorway,the Side-Angle-Side Postulatetells you that the triangles outlined at the right arecongruent. So, AB CD.Identifying Congruent TrianglesWould you use SSS or SAS to prove the triangles congruent? If there is not enoughinformation to prove the triangles congruent by SSS or SAS,write not enoughinformation. Explain your answer.What should youlook for first, sides or0 ----Aangles?Start with sides. If youhave three pairs ofcongruent sides, use SSS.If you have two pairs ofcongruent sides, lookUse SAS because two pairs offor a pair of congruentincluded angles.angles are congruent.There is not enough information; two pairs ofcorresponding sides are congruent, but oneof the angles is not the included angle.corresponding sides and their includedQ8Use SSS because three pairs ofcorresponding sides are congruent.Use SSS or SAS because all three pairs ofcorresponding sides and a pair of includedangles (the vertical angles) are congruent.Got It? 3. Would you use SSS or SAS to prove the triangles at theright congruent? Explain.PowerGeometry.comLesson 4-2TViangle Congruence by SSS and SAS229

Lesson CheckDo you know HOW?Do you UNDERSTAND?1. In APEN,name the angle that is included betweenthe given sides,b. NP and PEa. PE and EN5. Compare and Contrast How are the SSS Postulateand the SAS Postulate alike? How are they different?6. Error Analysis Your friend thinks that the trianglesshown below are congruent by SAS.Is your friendcorrect? Explain.2. In AHAT, between which sides is the given angleincluded?a. AHb. ATName the postulate you would use to prove thetriangles congruent.7. Reasoning A carpenter trims a triangular peakof a house with three 7-ft pieces of molding.Ihecarpenter uses 21 ft of molding to trim a secondtriangular peak. Are the two triangles formedcongruent? Explain.MATHEMATICALPractice and Problem-Solving ExercisesPractice @8. Developing Proof Copy and complete thePRACTICES See Problem 1.Mflow proof.Given: Jk LM,Jm LK-/Prove: AJKM ALMKJKsLMKJM LKGivena. ?KM KMb. ?c. ? d.SSS Given: IE GH,EE s HF,Proofp ismidpoint of GI10. Given: WZ ZS ' dWPf5?f Prove: AWZD ASDZProve: AEFl AHFGWH230Chapter 4 Congruent Triangles

See Problem 2.What other information,if any,do you need to prove the two trianglescongruent by SAS? Explain.12.11.WWLRMWould you use SSS or SAS to prove the triangles congruent? If there is notenough information to prove the triangles congruent by SSS or SAS,write not See Problem 3.enough information. Explain your answer.14.13.Apply @15. Think About a Plan You and a friend are cutting triangles out of felt for an artproject. You want all the triangles to be congruent. Your friend tells you that eachtriangle should have two 5-in. sides and a 40 angle. If you follow this rule, will allyour felt triangles be congruent? Explain. How can you use diagrams to help you? Which postulate, SSS or SAS, are you likely to apply to the given situation?17. Given: Xis the midpoint of AG and Ni?.1 . Given: BC DA, Z.CBD /LADBProofProof I Prove: AAWX LGBXAUse the Distance Formula to determine whether A ABC and ADEFarecongruent. Justify your answer.18. A(l,4), B{5,5), C(2, 2);D(-5, I), (-1,0),H-4.3)19. AC3, 8), B(8,12), C(10, 5);D(3, -1), (7,-7),H12,-2)20. A(2,9),5(2,4), C(5,4);D(l, -3), (1,2),F{-2,2)21. Writing List three real-life uses of congruent triangles. For each real-life use,describe why you think congruence is necessary.CPowerGeometry.corn{ Lesson 4-2 Triangle Congruence by SSS and SAS231

22. Sierpinski's Triangle Sierpinski's triangle is a famousgeometric pattern. To draw Sierpinski's triangle, start witha single triangle and connect the midpoints of tlie sidesto draw a smaller triangle. If you repeat this pattern overand over, you will form a figure like the one shown. Thisparticular figure started with an isosceles triangle. Are thetriangles outlined in red congruent? Explain.23. Constructions Use a straightedge to draw any triangleJKL.Then construct AMNP AJKL using thegiven postulate.a. SSSb. SASCan you prove the triangles congruent? Ifso, write the congruence statementand name the postulate you would use. If not, write notenough information andtell what other information you would need.24.25.26.NW27. Reasoning Suppose GH }K, HI KL, and LI LL.Is AGHI congruent toAJKLI Explain.M. Given: GK bisects LJGM, GJ GM29. Given: AE and BD bisect each other.Prove: AGJK AGMKProve: AACB AECDMM. Given: FG KL, FG KLProof Prove: AFGK s AKLF31. Given: AB 1 CM, AB 1 DB, CM DB,ProofM is the midpoint of ABProve: AJiMC AMBDD232Chapter 4 Congruent Triangles

A Challenge32. Given: HK LG,HP LJ, FG }KProof33. Given: AN AL, MN OL, NO s LMPtoof,Prove: MNProve: AFGH AJKLOLMFGN034. Reasoning Four sides of polygon ABCD are congruent, respectively, to the foursides of polygon EFGH.Are ABCD and EFGH congruent? Is a quadrilateral a rigidfigure? If not, what could you add to make it a rigid figure? Explain.rStandardized Test Prep35. What additional information do you need to prove thatAVWY AVWZ by SAS?CA W C AY AZce:: AWVY Awvzc vy36. The measures oftwo angles of a triangle are 43 and 38. What is themeasure of the third angle?XSlCh: 99CD 10037. Which method would you use to find the inverse ofa conditional statement?Short. Response iirCD Switch the hypothesis and conclusion.Cp Negate the conclusion only.CD Negate the hypothesis only.CD Negate both the hypothesis and conclusion.38. A segment has a midpoint at(1,1)and an endpoint at(—3,4). What are thecoordinates of tlie other endpoint ofthe segment? Show your work.rMixed Review4 See Lesson 4-1.ABCD EFGH. Name the angle or side that corresponds to each part.39. AA40. W41. BC42. AGWrite the converse of each statement. Determine whether the statement and its See Lesson 2-2.converse are true or false.43. If X 3, then 2x 6.44. If x 3,then 9.Get Ready! To prepare for Lesson 4-3 do Exercises 45 and 46.45. In AJHK, name the side that is included between A} and AH.See Lesson 4-2.46. In ANLM, name the angle that is included between NM and LN.cPowerGebmettv.comLesson 4-2 Triangle Congruence by SSS and SAS233

Oct 08, 2020 · You have two pairs of congruent sides. What else do you need? You need a third pair of congruent corresponding sides. Notice that the triangles share a common side, W. Postulate 4-1 Side-Side-Side (SSS) Postulate Postulate If the three sides of one triangle are congruent to the three sides