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Congruence of Triangle

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Congruence of Triangle

  1. 1. T- 1-855-694-8886Email- info@iTutor.comBy iTutor.com
  2. 2. Congruence of Triangles• Congruent triangles are triangles that have thesame size and shape. This means that thecorresponding sides are equal and the correspondingangles are equal• In the above diagrams, the corresponding sidesare a and d; b and e ; c and f.• The corresponding anglesare x and s; y and t; z and u.
  3. 3. Criteria for Congruence ofTrianglesThere are four rules to check for congruent triangles.SSS Rule (Side-Side-Side rule)SAS Rule (Side-Angle-Side rule)ASA Rule (Angle-Side-Angle Rule)AAS Rule (Angle-Angle-Side rule)Hypotenuse Leg Rule
  4. 4. ASA congruence ruleTwo triangles are congruent if two angles and theincluded side of one triangle are equal to two anglesand the included side of other triangleProof : We are given two triangles ABC and DEF inwhich: ∠ B = ∠ E, ∠ C = ∠ F and BC = EFTo prove that : Δ ABC ≅ Δ DEF,For proving the congruence of the two triangles seethat three cases arise.
  5. 5. ASA congruence ruleCase (i) : Let AB = DE in figureYou may observe thatAB = DE ……….(Assumed)∠ B = ∠ E …………(Given)BC = EF …………..(Given)So, Δ ABC ≅ Δ DEF…………….(By SAS rule)AB CDE Fll
  6. 6. AB CDE FllCase (ii) : Let if possible AB > DE.So, we can take a point P on ABsuch that PB = DE.Now consider Δ PBC and Δ DEF (see Fig.)ASA congruence rulePIn Δ PBC and Δ DEF,PB = DE ……………………………(By construction)∠ B = ∠ E ,BC = EF……………….. (Given)So, Δ PBC ≅ Δ DEF, by the SAS axiom for congruence.
  7. 7. AB CDE FllPASA congruence ruleSince the triangles are congruent, theircorresponding parts will be equal.So, ∠ PCB = ∠ DFEBut, given that ∠ ACB = ∠ DFESo, ∠ ACB = ∠ PCBThis is possible only if P coincides with A.or, BA = EDSo, Δ ABC ≅ Δ DEF …………………..(by SAS axiom)
  8. 8. Case (iii) : If AB < DE,we can choose a point M on DE such that ME = ABΔ ABC and Δ MEF (see Fig.)AB = ME ……………………………(By construction)∠ B = ∠ E ,BC = EF……………….. (Given)So, Δ ABC ≅ Δ MEF, by the SAS axiom for congruence.AB CDE FllMASA congruence rule
  9. 9. If Δ ABC ≅ Δ MEFthen corresponding parts will be equal.So, ∠ ACB = ∠ MFE, But ∠ ACB = ∠DFE…… (Given)so, ∠ ACB = ∠ MCBThis is possible only if M coincides with D.or, BA = ED• So, Δ ABC ≅ Δ DEF …………………..(by SAS axiom)ASA congruence ruleAB CDE FllM
  10. 10. • So all the three cases:-• Case (i) : AB = DE• Case (ii) : AB > DE• Case (iii) : AB < DE,We can see that Δ ABC ≅ Δ DEFProvedASA congruence ruleAB CDE Fll
  11. 11. SSS congruence ruleTwo triangles are congruent, if three sides of onetriangle are equal to the corresponding three sides ofthe other triangleGiven: Two Δ ABC and Δ DEF such that, AB = DE,BC = EF, and AC = DF.To Prove: To prove Δ ABC is congruent to Δ DEF.AB CDE F
  12. 12. SSS congruence ruleAB CDE FGConstruction: Let BC is the longest side.Draw EG such that, < FEG = < ABC,EG = AB. Join GF and GDProof: In Δ ABC & Δ GEFBC = EF ……………….(Given)AB = GE …………..(construction)< ABC = < FEG ……(Construction)Δ ABC ≅ Δ GEF< BAC = < EGF and AC = GFNow, AB = DE and AB = GEDE = GE……..Similarly,AC = DF and AC = GF,DF = GFIn Δ EGD, we haveDE = GE< EDG = < EGD ---------- (i)
  13. 13. AB CDE FGSSS congruence ruleIn Δ FGD, we haveDF = GF……….(ii)From (i) and (ii) we get,< EDF = < EGFBut, <EGF = <BAC,Therefore,< EDF = < BAC -----------(iii)In Δ ABC and Δ DEF, we have,AB = DE ………..(Given)< BAC = < EDGAC = DF ……..(Given)Therefore, by SAS congruence,Δ ABC ≅ Δ DEF
  14. 14. If in two right triangles the hypotenuse and one sideof one triangle are equal to the hypotenuse and oneside of the other triangle, then the two triangles arecongruent.Given: Two right angle TriangleABC and PQR whereAB = PQ and AC = PR,To Prove:Δ ABC ≅ Δ PQRProof:we Know that these Triangles are Right angleSo < B = <C ……………..(900)AB = PQ and AC = PR ------------(Given)Δ ABC ≅ Δ PQR -------------(by SAS)RHS congruence ruleABCPRQ
  15. 15. • Angles opposite to equal sides of an isoscelestriangle are equal.Given : An isosceles Δ ABC in whichAB = AC.To prove: ∠ B = ∠ CConstruction:Draw the bisector of ∠ Aand D be the point of intersection of this bisector of∠ A and BC.Proof: In Δ BAD and Δ CAD,AB = AC …………………..(Given)∠ BAD = ∠ CAD …………(By construction)Properties of a TriangleAB C
  16. 16. AD = AD ………………..(Common)So, Δ BAD ≅ Δ CAD ………(By SAS rule)So, ∠ ABD = ∠ ACD,since they arecorresponding angles of congruent triangles.So,∠ B = ∠ CProvedProperties of a TriangleAB C
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