Triangles

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Triangles

  1. 1. TrianglesTrianglesBy Eisa, Deva and Rajat
  2. 2. • Basic Proportionality Theorem• Similarity Criteria• Area Theorem• Pythagoras TheoremWhat will you learn?
  3. 3. • Basic Proportionality Theorem states that if aline is drawn parallel to one side of a triangleto intersect the other 2 points , the other 2sides are divided in the same ratio.• It was discovered by Thales , so also knownas Thales theorem.Basic Proportionality Theorem
  4. 4. ProvingtheThales’Theorem
  5. 5. Converse of the Thales’ TheoremIf a line divides any two sides of a triangle inthe same ratio, then the line is parallel to thethird side
  6. 6. ProvingtheconverseofThales’Theorem
  7. 7. Similarity CriteriaSimilarity CriteriasSSS ASA AA
  8. 8. • If in two triangles, corresponding angles areequal, then their corresponding sides are inthe same ratio (or proportion) and hence thetwo triangles are similar.• In Δ ABC and Δ DEF if ∠ A=∠ D, ∠ B= ∠E and ∠C =∠ F then Δ ABC ~ Δ DEF.AAA Similarity
  9. 9. • If in two triangles, sides of one triangle areproportional to (i.e., in the same ratio of ) thesides of the other triangle, then theircorresponding angles are equal and hencethe two triangles are similar.• In Δ ABC and Δ DEF if AB/DE =BC/EF =CA/FDthen Δ ABC ~ Δ DEF.SSS Similarity
  10. 10. • If one angle of a triangle is equal to one angleof the other triangle and the sides includingthese angles are proportional, then the twotriangles are similar.• In Δ ABC and Δ DEF if AB/DE =BC/EF and ∠B= ∠E then Δ ABC ~ Δ DEF.SAS Similarity
  11. 11. • The ratio of the areas of two similar trianglesis equal to the square of the ratio of theircorresponding sides• It proves that in the figuregiven belowArea Theorem
  12. 12. ProofofAreaTheorem
  13. 13. • If a perpendicular is drawn from the vertex of the rightangle of a right triangle to the hypotenuse then triangleson both sides of the perpendicular are similar to thewhole triangle and to each other• In a right triangle, the square of the hypotenuse is equalto the sum of the squares of the other two sides.• In a right triangle if a and b are the lengths of the legs andc is the length of hypotenuse, then a² + b² = c².• It states Hypotenuse² = Base² + Altitude².• A scientist named Pythagoras discovered the theorem,hence came to be known as Pythagoras Theorem.Pythagoras Theorem
  14. 14. Pythagoras Theorem
  15. 15. ProofofPythagorasTheorem
  16. 16. In a triangle, if square of one side is equal tothe sum of the squares of the other twosides, then the angle opposite the first side is aright angle.Converse of Pythagoras Theorem
  17. 17. ProofofConverseofPythagorasTheorem
  18. 18. • Two figures having the same shape but not necessarily the samesize are called similar figures.• All the congruent figures are similar but the converse is nottrue.• Two polygons of the same number of sides are similar, if (i)their corresponding angles are equal and (ii) theircorresponding sides are in the same ratio (i.e., proportion).• If a line is drawn parallel to one side of a triangle to intersectthe other two sides in distinct points, then the other two sidesare divided in the same ratio.• If a line divides any two sides of a triangle in the same ratio,then the line is parallel to the third side.Summary
  19. 19. • If in two triangles, corresponding angles are equal, then theircorresponding sides are in the same ratio and hence the two trianglesare similar (AAA similarity criterion).• If in two triangles, two angles of one triangle are respectively equal tothe two angles of the other triangle, then the two triangles are similar(AA similarity criterion).• If in two triangles, corresponding sides are in the same ratio, then theircorresponding angles are equal and hence the triangles are similar (SSSsimilarity criterion).• If one angle of a triangle is equal to one angle of another triangle andthe sides including these angles are in the same ratio (proportional),then the triangles are similar (SAS similarity criterion).• The ratio of the areas of two similar triangles is equal to the square ofthe ratio of their corresponding sides.Summary
  20. 20. • If a perpendicular is drawn from the vertex of the right angle of a righttriangle to the hypotenuse, then the triangles on both sides of theperpendicular are similar to the whole triangle and also to each other.• In a right triangle, the square of the hypotenuse is equal to the sum ofthe squares of the other two sides (Pythagoras Theorem).• If in a triangle, square of one side is equal to the sum of the squares ofthe other two sides, then the angle opposite the first side is a right angle.Summary
  21. 21. Thank you

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