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- 3. We know that a closed figure formed by three intersecting lines is called a triangle(‘Tri’ means ‘three’).A triangle has three sides, three angles and three vertices. For e.g.-in Triangle ABC, denoted as ∆ABC AB,BC,CA are the three sides, ∠A,∠B,∠C are three angles and A,B,C are three vertices. A B C
- 4. OBJECTIVES IN THIS LESSON • INEQUALITIES IN A TRIANGLE. 2 • STATE THE CRITERIA FOR THE CONGRUENCE OF TWO TRIANGLES. 3 • SOME PROPERTIES OF A TRIANGLE. 4 •DEFINE THE CONGRUENCE OF TRIANGLE. 1
- 5. DEFINING THE CONGRUENCE OF TRIANGLE:- Let us take ∆ABC and ∆XYZ such that corresponding angles are equal and corresponding sides areequal :- A B C X Y Z CORRESPONDING PARTS ∠A=∠X ∠B=∠Y ∠C=∠Z AB=XY BC=YZ AC=XZ
- 6. Now we see that sides of ∆ABC coincides with sides of ∆XYZ. B C A X Y Z SO WE GET THAT TWO TRIANGLES ARE CONGRUENT, IF ALL THE SIDES AND ALL THE ANGLES OF ONE TRIANGLE ARE EQUAL TO THE CORRESPONDING SIDES AND ANGLES OF THE OTHER TRIANGLE. Here, ∆ABC ≅ ∆XYZ
- 7. This also means that:- A corresponds to X B corresponds to Y C corresponds to Z Forany twocongruent triangles the corresponding parts areequal and are termed as:- CPCT – Corresponding Parts of Congruent Triangles
- 8. CRITERIAS FOR CONGRUENCE OF TWO TRIANGLES SAS(side-angle-side) congruence • Twotrianglesarecongruent if twosidesand the included angleof one triangleare equal to the twosidesand the included angleof othertriangle. ASA(angle-side-angle) congruence • Twotrianglesarecongruent if twoanglesand the included sideof one triangleare equal to twoanglesand the included sideof othertriangle. AAS(angle-angle-side) congruence • Twotrianglesarecongruent if any two pairs of angleand one pairof corresponding sidesareequal. SSS(side-side-side) congruence • If three sidesof one triangleareequal to the three sidesof another triangle, then the two trianglesarecongruent. RHS(right angle-hypotenuse-side) congruence • If in two right-angled triangles the hypotenuseand onesideof one triangleareequal to the hypotenuseand onesideof theother triangle, then the two trianglesarecongruent.
- 9. A B C P Q R S(1) AC = PQ A(2) ∠C = ∠R S(3) BC = QR Now If, Then ∆ABC ≅ ∆PQR (by SAS congruence)
- 10. A B C D E F Now If, A(1) ∠BAC = ∠EDF S(2) AC = DF A(3) ∠ACB = ∠DFE Then ∆ABC ≅ ∆DEF (by ASA congruence)
- 11. A B C P Q R Now If, A(1) ∠BAC = ∠QPR A(2) ∠CBA = ∠RQP S(3) BC = QR Then ∆ABC ≅ ∆PQR (by AAS congruence)
- 12. Now If, S(1) AB = PQ S(2) BC = QR S(3) CA = RP A B C P Q R Then ∆ABC ≅ ∆PQR (by SSS congruence)
- 13. Now If, R(1) ∠ABC = ∠DEF = 90° H(2) AC = DF S(3) BC = EF A B C D E F Then ∆ABC ≅ ∆DEF (by RHS congruence)
- 14. PROPERTIES OF TRIANGLE A B C A Triangle in which two sides are equal in length is called ISOSCELES TRIANGLE. So, ∆ABC isa isosceles trianglewith AB = BC.
- 15. Angles opposite to equal sides of an isosceles triangle are equal. B C A Here, ∠ABC = ∠ ACB
- 16. The sides opposite to equal angles of a triangle are equal. C B A Here, AB = AC
- 17. Theorem on inequalities in a triangle If two sides of a triangleare unequal, theangleopposite to the longer side is larger( orgreater) 10 9 8 Here, bycomparing wewill get that- Angleopposite to the longer side(10) is greater(i.e. 90°)
- 18. In any triangle, the side opposite to the longer angle is longer. 10 9 8 Here, bycomparing we will get that- Side(i.e. 10) opposite to longerangle (90°) is longer.
- 19. The sum of any two side of a triangle is greater than the third side. 10 9 8 Here bycomparing weget- 9+8>10 8+10>9 10+9>8 So, sum of any two sides isgreater than the third side.
- 20. SUMMARY 1.Twofiguresarecongruent, if theyareof the same shape and size. 2.If twosides and the included angleof one triangle is equal to the twosides and the included angle then the two trianglesarecongruent(by SAS). 3.If two angles and the included side of one triangle are equal to the two angles and the included side of othertriangle then the two trianglesarecongruent( by ASA). 4.If two angles and the one side of one triangle is equal to the two angles and the corresponding side of othertriangle then the two trianglesarecongruent(byAAS). 5.If three sides of a triangle is equal to the three sides of other triangle then the twotrianglesarecongruent(by SSS). 6.If in tworight-angled triangle, hypotenuseoneside of the triangleareequal to the hypotenuseand one side of the other triangle then the two triangle are congruent.(byRHS) 7.Angles opposite to equal sides of a triangle are equal. 8.Sides opposite to equal angles of a triangle are equal. 9.Eachangleof equilateral triangleare 60° 10.Ina triangle, anglesopposite tothe longerside is larger 11.Ina triangle, sideopposite to the largerangle is longer. 12.Sum of any twosides of triangle is greaterthan the third side.